Perturbations of linear quasiperiodic system
L. Hakan Eliasson
160
KAMpersistence of finitegap solutions
Sergei B...
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Perturbations of linear quasiperiodic system
L. Hakan Eliasson
160
KAMpersistence of finitegap solutions
Sergei B. Kuksin
61123
Analytic linearization of circle diffeomorphisms
JeanChristophe Yoccoz
125173
Some open problems related to small divisors
Stefano Marmi and JeanChristophe Yoccoz
175191
Perturbations of linear quasiperiodic system L. H. Eliasson
1
Introduction
Existence of both Floquet and l2 solutions of linear quasiperiodic skewproducts can be formulated in terms of linear operators on l2 (Z), i.e. ∞dimensional matrices. In the perturbative regime these matrices are perturbations of diagonal matrices and the problem is to diagonalize them completely or partially, i.e. to show that they have some point spectrum. The unperturbed matrices have a dense point spectrum so that their eigenvalues are, up to any order of approximation, of inﬁnite multiplicity, which is a very delicate situation to perturb. For matrices with strong decay of the matrix elements oﬀ the diagonal this diﬃculty can be overcome if the eigenvectors are suﬃciently well clustering. One way to handle this is to control the almost multiplicites of the eigenvalues. The eigenvalues are given by functions of one or several parameters and in order to control the almost multiplicities it is necessary that these functions are not too ﬂat. Such a condition is delicate to verify since derivatives of eigenvalues of a matrix behave very bad under perturbations of the matrix. Derivatives of eigenvalues of matrices are therefore replaced by derivatives of resultants of matrices – an object which behaves better under perturbations. If the parameter space is onedimensional and if the quasiperiodic frequencies satisfy some Diophantine condition, then it turns out that this control of the derivatives of eigenvalues, in terms of the resultants, is not only necessary but also suﬃcient for the control of the almost multiplicities. If the parameter space is higherdimensional this control is more diﬃcult to achieve and not yet well understood. 1.1
Examples
Discrete Schr¨ odinger equations in one dimension. In strong coupling regime this equation takes the form −ε(un+1 + un−1 ) + V (θ + nω)un = Eun ,
n ∈ Z,
(1.1)
where V is a realvalued piecewise smooth function on the ddimensional torus Td = (R/2πZ)d and ω is a vector in Rd . The constant ε is a way to measure the size of the potential V and it is assumed to be small, i.e. a large potential. E is a real parameter.
L.H. Eliasson et al.: LNM 1784, S. Marmi and J.C. Yoccoz (Eds.), pp. 1–60, 2002. c SpringerVerlag Berlin Heidelberg 2002
2
L. H. Eliasson
When ε = 0 (1.1) have for each m ∈ Z the solution um ∈ l2 (Z),
m um n = en
if we choose E = V (θ + mω) – em n is 1 when n = m and 0 otherwise. A solution u ∈ l2 (Z) is an eigenvector of the lefthand side of (1.1) interpreted as an operator on l2 (Z). Represented in the standard basis for l2 (Z) it becomes an ∞dimensional matrix .. . .. . 0 −1 + ε V (θ + nω) −1 0 −1 .. −1 0 . .. . over Z. This matrix has the following properties: * dependence of parameters θ ∈ Td ; * covariance with respect to the Zaction on Td θ → θ + nω – covariance means that we obtain any row/column, and hence the whole matrix, from one row/column simply by shifting the parameter θ by the group action; * dense spectrum of the diagonal part; * domination of diagonal part when ε is small; * strong decay oﬀ the diagonal. We can also notice that the matrix is symmetric and that the diagonal part is determined by the potential V while the perturbation is independent of the potential. A natural question is if this matrix has a basis of eigenvectors in l2 (Z), in which case it can be diagonalized, or if it has any eigenvectors at all. Such eigenvectors will give solutions to (1.1) which are decaying both in forward and backward time. The existence of eigenvectors for small but positive ε is a delicate question because in the unperturbed case all eigenvalues are essentially of inﬁnite multiplicity. Moreover there are numerous examples when there are no eigenvectors: if V is constant there are no eigenvectors since the operator un → −ε(un+1 + un−1 ) is absolutely continuous on l2 (Z); if the potential is periodic, i.e. ω ∈ 2πQd , there are no eigenvectors; if the potential is nonperiodic there are examples without eigenvectors when ω is Liouville. In weak coupling we consider the equation in the form −(un+1 + un−1 ) + εV (θ + nω)un = Eun ,
n ∈ Z.
(1.2)
Perturbations of linear quasiperiodic system
3
When ε = 0 (1.2) has, for each ξ ∈ T, extended solutions uξ ∈ l∞ (Z),
uξn = einξ ,
for E = −2 cos(ξ). When ε is small it is natural to look for extended solutions of the form uξn = einξ U (θ + nω), where U : Td → C. Such solutions are known as Floquet solutions or Bloch waves. The equation for U then becomes −(eiξ U (θ + ω) + e−iξ U (θ − ω)) + εV (θ)U (θ) = EU (θ) which, when written in Fourier coeﬃcients, gives the matrix .. . .. 0 Vˆ (k − l) . − ε 2 cos(ξ+ < k, ω >) .. Vˆ (l − k) 0 .
..
.
over Zd . This matrix has the properties: * dependence of parameters ξ ∈ T; * covariance with respect to the Zd action on T ξ → ξ+ < k, ω > . It has like (1.1) a dense spectrum of the diagonal part, domination of diagonal part when ε is small and decay oﬀ the diagonal (depending on the smoothness of V ). In this case however, the matrix is complex and Hermitian, the diagonal part is ﬁxed = 2 cos(α) and the perturbation depends on the potential function V . Finding Floquet solutions for (1.2) now amounts to ﬁnding eigenvectors of this matrix. This is a delicate matter which is known to depends on arithmetical properties of ω and requires strong smoothness of V which is reﬂected in strong decay oﬀ the diagonal of the matrix. Floquet solutions are related to the absolutely continuous spectrum and there are examples of discontinuous potentials with only singular spectrum. In strong coupling we can consider generalizations of the form −ε(∆θ u)n + V (θ + nω)un = Eun ,
n ∈ Z,
where the diﬀerence operator is allowed to take a more general form (∆θ u)n =
N j=−N
bj (θ + (n + j)ω)un+j .
(1.1 )
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L. H. Eliasson
It could even be inﬁnite with suﬃciently strong decay. This equation will now give rise to a perturbation problem with the same diagonal part as (1.1) but with a perturbation that depends on the bj ’s. It is typically not symmetric and it is therefore not reasonable to try to diagonalize it for small ε. But block diagonalization and construction of eigenvectors still make sense. In weak coupling we can consider equation −(∆u)n + εV (θ + nω)un = Eun ,
n ∈ Z,
(1.2 )
with a constant diﬀerence operator of the more general form N
(∆u)n =
bj un+j .
j=−N
It can also be inﬁnite with suﬃciently strong decay. It will now give rise to a perturbation problem of the same type as (1.2) but with diagonal part determined by the function N bj eijξ . j=−N
Discrete Linear SkewProducts. We consider a weakly perturbed skewproduct of the form Xn+1 − (A + εB(θ + nω))Xn = EXn ,
n ∈ Z,
(1.3)
where A ∈ Gl(N, R), B : Td → gl(N, R). When ε = 0 we have the Floquet solutions Xn = einξ X0 ,
AX0 = aX0 ,
for E = eiξ − a, ξ ∈ T. For ε = 0 we look for Floquet solutions Xn = einξ Y (θ + nω) with Y : Td → CN . The equation for Y then becomes eiξ Y (θ + ω) − (A + εB(θ))Y (θ) = EY (θ) which, when expressed in Fourier coeﬃcients, becomes the matrix .. . .. . ˆ − l) 0 B(k − ε ei(ξ+) I − A ˆ − k) B(l 0 .. . over Zd . The properties of the matrix are:
..
.
Perturbations of linear quasiperiodic system
5
* dependence of parameters ξ ∈ T; * covariance with respect to the Zd action on T ξ → ξ+ < k, ω > . The matrix has dense point spectrum of diagonal part, domination of diagonal part when ε is small and decay oﬀ the diagonal (depending on the smoothness of B). Notice that here the unperturbed part is only block diagonal and its spectrum is determined by the functions (eξ − aj ) where aj runs over the eigenvalues of A – a multilevel operator. Finding Floquet solutions for (1.3) now amounts to ﬁnding eigenvectors of this matrix. We shall give two example of skewproducts which can be considered as strongly perturbed. The ﬁrst example is an obvious generalization of the Schr¨ odinger equation: −ε(Xn+1 + Xn−1 ) + B(θ + nω)Xn = EXn ,
n ∈ Z,
(1.4)
where B : Td → gl(N, R) is a symmetric matrix. This now gives rise to a multilevel version of (1.1). The second example arises from the continuous Schr¨odinger equation in strong coupling −εu (t) + V (θ + tω)u(t) = Eu(t),
t ∈ R.
(1.5)
For ε = 0 the operator to the left becomes multiplication by V (θ + tω) which has purely continuous spectrum and no eigenvalues. Hence this is an even more complicated object to perturb from. Instead we consider the corresponding dynamical system u v
=
0 1 ε (V
u 1
(θ + tω) − E) 0
v
,
θ = θ + ω,
and its timetmap (Φt (θ; E, ε), θ + ω). Let ω = (ω , 1) and θ = (θ , θd ). Then the evolution of the system is described by un+1 an bn un un = =: Φ1 ((θ + nω , 0); E, ε) , vn+1 cn dn vn vn where an dn − bn cn = 1. From this we deduce the second order equation bn−1 un+1 + bn un−1 − W (θ + nω )un = 0, where W (θ ) = a0 (θ )b0 (θ − ω ) + d(θ − ω )b(θ ).
(1.6)
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L. H. Eliasson
In matrix formulation the righthand side of (1.6) becomes .. . .. 0 bn−2 . + bn 0 bn−1 W (θ + nω ) .. bn+1 0 .
..
. .
which is a matrix over Z with properties: * dependence of parameters θ ∈ Td−1 ; * covariance with respect to the Zaction on Td−1 θ → θ + nω . The diagonal part has dense spectrum and the matrix has strong decay oﬀ the diagonal. There is some domination of the diagonal part if ε is small and if E is close to inf V but it is less explicit than in (1.1). Both the diagonal and the nondiagonal part depend on the potential V . Since the matrix is neither symmetric nor Hermitian we cannot hope to diagonalize it, but it may still be block diagonalizable and have a pure point spectrum with ﬁnite multiplicity. If 0 is an eigenvalue for a given parameter value E, then the corresponding eigenvector will give a solution of (1.5) which is decaying both in forward and backward time giving some point spectrum for that operator. Higherdimensional Schr¨ odinger equations. We can consider the Schr¨odinger equation in, say, two dimensions in strong coupling −ε(∆u)n + V (θ1 + n1 ω1 , θ2 + n2 ω2 )un = Eun ,
n ∈ Z2 ,
(1.7)
where ∆ is the nearest neighbor diﬀerence operator (∆u)n = um , m−n=1
V is a real valued piecewise smooth function on the product of two tori Td ×Td and ω = (ω1 , ω2 ) is a vector in Rd × Rd . A solution u ∈ l2 (Z2 ) is an eigenvector of a perturbation of the matrix .. . V (θ1 + n1 ω1 , θ2 + n2 ω2 ) .. . over Z2 . This matrix has the following properties:
Perturbations of linear quasiperiodic system
7
* dependence of parameters θ ∈ T2d ; * covariance with respect to the Z2 action on T2d θ = (θ1 , θ2 ) → (θ1 + n1 ω1 , θ2 + n2 ω2 ). The diagonal part has dense spectrum and dominates the matrix when ε is small, and the decay oﬀ the diagonal is strong. In weak coupling −(∆u)n + εV (θ1 + n1 ω1 , θ2 + n2 ω2 )un = Eun ,
n ∈ Z2 ,
(1.8)
the existence of Floquet solutions un = ei(n1 ξ1 +n2 ξ2 ) U (θ1 + n1 ω1 , θ2 + n2 ω2 ) gives rise to an eigenvalue problem in the same way as (1.2). 1.2
Two basic diﬃculties
The fundamental problem in the perturbation theory of these matrices is that all eigenvalues have inﬁnite multiplicity up to any order of approximation. More precisely, we say that two eigenvalues E m (θ) and E n (θ) are ρalmost multiple if  E m (θ) − E n (θ)  ≤ ρ. Then the ρalmost multiplicity is inﬁnite for any ρ = 0. This is not only a technical obstacle since for example the matrix derived from equation (1.1) does not have any eigenvectors for any ε = 0 if V is constant. Inﬁnite almostmultiplicity can be handled if the matrix has strong decay oﬀ the diagonal and if the corresponding almostmultiple eigenvectors {umi } cluster. By this we mean that the range of all the (umi )’s, i.e. the inﬁnite subset i ∪i {n ∈ Z : um n = 0} collects into ﬁnite clusters in Z. This clustering of almostmultiple eigenvectors is crucial for the possible diagonalization of the perturbed matrix. It should be measured by the separation and extension of the clusters – which are related to the decay of the matrix oﬀ the diagonal – and by the the cardinality of the clusters. The ﬁrst diﬃculty is therefore to get good clustering of eigenvectors. An essential role in this is played by the strong decay of the perturbation oﬀ the diagonal and by the Diophantine properties of the frequency vector ω. A vector ω ∈ Rd is said to be Diophantine with parameters κ, τ > 0 if inf < n, ω > −2πm  ≥
m∈Z
κ ,  n τ
∀n ∈ Zd \ 0.
(1.9)
8
L. H. Eliasson
Control of the almost multipicities and clustering. In the problem (1.1) the eigenvalues E m (θ) and the eigenvectors um (θ) are E m (θ) = V (θ + mω)
m and um n = en .
Which are the eigenvalues that are ρalmost multiple to E m (θ) and how many are there in any ﬁnite subset of Z? In order to investigate this question, consider the set ∆ = {x ∈ T : E m (θ + x) − E m (θ) ≤ ρ}. If V is smooth, measured by β and γ,  V C k ≤ β(k!)2 γ k
∀k ≥ 0,
and if V satisﬁes the transversality condition, measured by σ and s, max  ∂xk (V (θ + x) − V (θ))  ≥ σ
0≤k≤s
∀θ, x,
then ∆ (see Lemma B1) is a union of intervals ∪∆j with a ﬁnite number ≤ µ (independent of ε) of components, and each component is contained in an interval with length 1 ≤ const ρ s – when V (θ) = cos(θ) the numbers µ and s are both 2. If there are more than µ + 1 eigenvalues E nj (θ), j = 1, 2, . . . , µ + 1, that are ρalmost multiple to E m (θ), then it follows that θ + ni ω and θ + nj ω belong to one component, ∆1 say, for at least two diﬀerent i, j ≤ µ + 1. What controls the return of a translation θ → θ + ω to an interval is the Diophantine condition (1.9). Under this condition it follows that 1 1 1  ni − nj ≥ constκ τ ( ) τ s , ρ
which means that when ρ is small then the distance between the sites ni and nj is very large. Consider now a sequence of eigenvalues E m (θ) = E n1 (θ), E n2 (θ), . . . , nµ+1 E (θ) such that any two consecutive pairs are ρclose. Then any two pairs are (µ + 1)ρclose and hence it must hold that max
1≤i,j≤µ+1
1
 ni − nj ≥ ν = constκ τ (
1 1 ) τs . (µ + 1)ρ
This implies that the ranges of the corresponding eigenvectors uni , i.e. the set of indices or sites {ni }, collects into clusters in Z containing at most µ many sites. Any two clusters are separated by at least ν many sites, and each cluster extends over at most λ = µν sites.
Perturbations of linear quasiperiodic system
9
This simple argument which controls the clustering of the eigenvectors does not depend on the dimension of the lattice Z, but it does depend crucially on the dimension of the parameter space {θ}. If we consider V (θ) = cos(θ1 ) + cos(θ2 ), deﬁned on the 2dimensional torus, with ω = (ω1 , ω2 ), then the corresponding set ∆ is now a subset of T2 and its components ∆i are simply connected subsets of T2 whose shape depend on the potential V . A Diophantine condition on ω does not alone control the return of a translation to an open connected set – the return also depend on the geometry of the sets ∆i . Of course the situation is the same if we consider a truly higherdimensional problem like (1.7) or (1.8). This is why the higherdimensional case is more diﬃcult and why our results in this paper are restricted to matrices which are covariant with respect to a group action on the onedimensional space T1 . Propagation of smoothness and transversality. The second diﬃculty is that the diagonalization is a small divisor problem which requires a quadratic iteration of KAMtype. It is therefore not enough to control the almostmultiplicities of the unperturbed matrix but they must be controlled also for nearby matrices – if not for all so at least for a suﬃciently large class of nearby matrices. This class of matrices is the normal form matrices: their elements have exponential decay oﬀ the diagonal measured by a parameter α; their smoothness is measured by β, γ; they are not smooth on the whole parameter space but only piecewise smooth over a partition P. They will have a pure point spectrum and will be characterized by a certain block structure – measured by λ, µ, ν, ρ – which describes their eigenvectors. However, they will not be diagonal nor block diagonal in general. Since derivatives of eigenvalues behave very bad under perturbations we cannot hope that the transversality condition of the eigenvalues of the unperturbed matrix propagates to the eigenvalues of the nearby normal form matrices. But instead of comparing two eigenvalues we can compare the whole spectrum of two submatrices truncated over blocks. This is done by the resultant and what we need is a transversality condition of the resultants of submatrices truncated over blocks – transversality of blocks. In distinction to eigenvalues, derivatives of resultants behave nicely under perturbations and we are able to show that this transversality condition of a normal form matrix propagates to nearby normal form matrices if their block structures are compatible in a certain sense. 1.3
Results and References
Some attempts to prove pure point spectrum for the discrete quasiperiodic Schr¨ odinger equation (1.1) on a onedimensional torus in strong coupling were made already in the early 80’s [1,2] but the breakthrough came in the late 80’s with the work of Sinai [3] and of Fr¨ ohlich, Spencer and Wittver [4]
10
L. H. Eliasson
on a “cosine”like potential. The most general result today is [5]. The present article provides a diﬀerent presentation of the result in [5], together with generalizations and variants. Similar results have also been obtained in [6] by diﬀerent methods. Discrete equations (1.1 ) were treated in [5,7]. In this paper we generalize the these results still a bit more. It should be noted the we only discuss the perturbation theory for smooth systems with Diophantine frequencies. There are numerous results for systems that are nonsmooth or have nonDiophantine frequencies. For example, the results referred to above about Liouville frequencies and discontinuous potentials can be found in [8,9]. The results on weak coupling (1.2) are older. Dinaburg and Sinai [10] constructed Floquet solutions for the continuous Schr¨ odinger in order to get some absolutely continuous spectrum (see also [11]). These results were then extended in [12] and the proof that there is such a solution for almost all quasimomentum was given in [13]. The work [13] even proves that the oneparameter family (1.2) have Floquet solutions for a.e. value of the spectral parameter E. All these results were proved for the continuous equation but the adaptation to the discrete case is likely to be quite straight forward. The results of [10] was for example carried over to the discrete case in [14]. The results [10–14] were proven by ODEmethods. An operator approach like the one in this paper were used in [15] and later also in [16]. For more general weakly perturbed skewproducts (1.3) results have been obtained in [17–19], also with the use ODEmethods. (See also the short survey [20].) Some further results are obtained in this paper. The continuous Schr¨ odinger in strong coupling (1.5) has only been treated in one particular case [4]: −εu (t) + (cos(θ1 + t) + cos(θ2 + tω))u(t) = Eu(t). This equation has, for ε small enough, a pure point spectrum close to the bottom of the spectrum. (From [13] it follows that it has purely absolutely continuous spectrum with Floquet solutions for suﬃciently large E.) Other strongly perturbed skewproducts have been studied in [21] from the point of view of Lyapunov exponents. For equation (1.1) on a higherdimensional torus there are two papers [22,23] which, in particular, stresses the importance of “external” parameters in these problems. In truly higher dimension essentially nothing is proven. For the continuous analogue of (1.8) some results, also with the use of “external” parameters, have been announced [24,25]. The diﬃculties met in higher dimensions are related to those in other problems on inﬁnitedimensional matrices [26]. In this paper we only discuss discrete quasiperiodic skewproducts. The continuous case is both similar and diﬀerent. The construction of Floquet solution does give rise to a similar perturbation problem. The diﬀerence is that the parameter space is the Euclidean space R, instead of the compact
Perturbations of linear quasiperiodic system
11
circle T, and that the spectrum of the diagonal part is unbounded. This is likely to be a minor point and we believe that all the results in weak coupling carry over from discrete to continuous skewproducts. The construction of l2 solutions of continuous skewproducts is, however, much more diﬃcult since it does not give rise to such a nice perturbation problem as in the discrete case. 1.4
Description of the paper
In Section 1 we introduce some notations. In Section 2 we deﬁne the normal form matrices which are a sort of generalized block matrices. They depend smoothly on some parameters and are covariant with respect to a group action. The smoothness is only piecewise in a way we explain. In Section 3 we show that we can conjugate the normal form matrices to block diagonal form, but there is a prize to pay – the smoothness properties of the conjugated matrix is less good. In Section 4 we show that we can conjugate a perturbation of a normal form matrix to a new normal form matrix with a much smaller perturbation. This should be the starting point for an iteration of KAMtype. In Section 5 we discuss under what conditions this procedure can be iterated and the role of clustering of the blocks. Up to this point neither the smoothness nor the group action play any role whatsoever. The clustering property is related to almost multiplicities of the eigenvalues. These multiplicities are related to estimates of resultants which can be obtained from a transversality property. In Section 6 we discuss when and how this transversality property of the resultants can be transferred to nearby normal form matrices. In Section 7 we specialize to a quasiperiodic group action on the onedimensional torus, i.e. to a linear ergodic action on T, submitted to a Diophantine condition. In this case the transversality property of the resultants will be suﬃcient to control the almost multiplicities of the eigenvalues and hence the clustering of the blocks. This will give us a perturbation theorem, Theorem 12, of a quite general type. In Section 8 we discuss applications of Theorem 12 to some of the onedimensional problems we have discussed above. In an appendix we include basic ﬁnitedimensional results from analysis and linear algebra. There is some diﬀerence, besides presentation, from the approach in [5]. We work with Gevrey functions – the particular Gevrey class has no importance – but instead of using the possibility to smoothly truncate Gevrey function as in [5] we use here discontinuous cutoﬀs. The disadvantage is that we have to work with piecewise smooth functions, which is a little awkward in relation to the covariance property, and that we must control the size of the pieces. The advantage is that the particular group action does not intervene before the ﬁnal step of the proof.
12
1.5
L. H. Eliasson
Notations
Let L be a standard lattice Z, Z2 , Z3 , . . . and denote its elements, called indices or sites, by a, b, c, . . . . If Ω, Ω ⊂ L, then we let dist(Ω, Ω ) =
min
a∈Ω,b∈Ω
 a − b ,
where   denotes the Euclidean distance. We let {eb : b ∈ L} denote the standard basis of l2 (L), deﬁned by eba = 1 or 0 depending on if a = b or not. If Ω is a subset of L, we denote by CΩ the subspace of l2 (L) spanned by {ea : a ∈ Ω}. A subspace Λ of dimension k of l2 (L) comes equipped with an ONframe, i.e. Λ is a linear mapping Λ : Ck → l2 (L) such that
Λ∗ Λ = IdCk ΛΛ∗ = ⊥ −projection of l2 (L) onto Λ.
The range of Λ is the set of sites that Λ occupies, i.e. the set of all a ∈ L such that (ea )∗ Λ = 0. A block for Λ is a subset Ω ⊂ L that contains the range of Λ. If D : l2 (L) → l2 (L) is a bounded operator we identify it with an ∞dimensional matrix via the standard basis: Dab = (ea )∗ Deb , the matrix element at row a and column b. By a matrix on L we shall always understand such a representation of a bounded operator. The truncation at ˜ deﬁned by D ˜ b = Db or = 0 distance N oﬀ the diagonal of D is the matrix D a a depending on if  b − a ≤ N or not. If Λ is a subspace we let DΛ = Λ∗ DΛ and if Ω ⊂ L we denote by DΩ the matrix DCΩ . A partition of a manifold X is a (locally ﬁnite) collection of open subsets – pieces – P such that ∪Y ∈P Y is of full measure in X. If P and Q are two such decomposition and if T is a homeomorphism on X, then
P ∨ Q = {Y ∩ Z : Y ∈ P, Z ∈ Q} T (P) = {T (Y ) : Y ∈ P}. Td denotes the ddimensional torus (R/2πZ)d and its distance is denoted by . We denote by  u  the Euclidean norm if u is a vector and the operator norm if u is a linear operator. For a smooth function u (possibly vector or matrix valued) deﬁned in an open box V in Rd we let  u C k be sup sup  0≤l≤k x∈V
1 l ∂ uj (x) , (l!)2
k ∈ Nd .
Perturbations of linear quasiperiodic system
13
u is said to be of Gevrey class G2 if  u C k ≤ βγ k
∀k ≥ 0.
Smooth function will always be assumed to be G2 unless otherwise speciﬁed – that we have chosen this particular Gevrey class will be of no importance. A function is said to be smooth on P if it is smooth on (a neighborhood of the closure of) each piece of P.
2 2.1
Covariance and normal form matrices Covariance
Let L be a lattice Z, Z2 , . . . and let X be a a torus T, T2 , . . . . On the torus X we have an Laction T : L × X → X, and we denote by Ta the mapping x → T (a, x), a ∈ L, x ∈ X. For an element b of L, deﬁne τb : l2 (L) → l2 (L) by (τb f )a = fa−b . Clearly τb is unitary and τb∗ = τb−1 = τ−b . Deﬁnition. A matrix D : l2 (L) × X → l2 (L) is covariant with respect to the group action T if τc∗ D(x)τc = D(Tc (x))
∀ c ∈ L, x ∈ X.
A covariant matrix D(x) is pure point if, for almost all x, there exists an eigenvector q(x) such that {q a (x) =: τa q(Ta (x)) : a ∈ L} is a basis for l2 (L). Ω(x) ⊂ L is a block for q(x) if Ω(x) ⊃ {a ∈ L : (ea )∗ q(x) = 0}. If q(x) is an eigenvector with eigenvalue E(x) and block Ω(x), q b (x) = τb q(Tb (x)) is an eigenvector with eigenvalue E b (x) = E(Tb (x)) and block Ωb (x) = Ω(Tb (x)) − b. In terms of the matrix elements, covariance means that b+c (x) = Dab (Tc x) Da+c
∀a, b, c ∈ L, x ∈ X.
More generally, we say that D(x) is pure point if q(x) is a generalized eigenvector with ﬁnite multiplicity. A block for q(x) is then a block for the smallest invariant space containing q(x).
14
L. H. Eliasson
Even more generally we can consider multilevel operators D : l2 (L) ⊗ CN × X → l2 (L) ⊗ CN . On l2 (L) ⊗ CN we have the standard basis {eb,j : b ∈ L, j = 1, . . . , N } deﬁned by eb,j a,i = 1 or 0 depending on if (a, i) = (b, j) or not. For an element b ∈ L, we deﬁne τb : l2 (L) ⊗ CN → l2 (L) ⊗ CN by (τb f )a,i = fa−b,i ,
i = 1, . . . , N.
The covariance of D with respect to the Laction T is now deﬁned in the same way as above. D(x) is pure point if l2 (L) ⊗ CN has a basis {q b,j (x) : a ∈ L, j = 1, . . . N } of eigenvectors of D(x). A block for q j is a subset Ω(x) ⊂ L × {1, . . . , N } such that Ω(x) ⊃ {(a, i) ∈ L × {1, . . . , N } : (ea,i )∗ q j (x) = 0}. We can also consider in the multilevel case generalized eigenvectors of ﬁnite multiplicity q j (x). 2.2
Normal Form Matrices
We shall consider matrices D : l2 (L) × X → l2 (L) which are covariant with respect to an Laction T :L×X →X and which satisfy the following conditions. Exponential decay oﬀ the diagonal.  Dab C 0 ≤ βe−αb−a .
(2.1)
Smoothness. The components of D are piecewise smooth and satisfy  Dab C k ≤ βe−αb−a γ k
∀k ≥ 1.
(2.2)
D(x) is pure point with a (possibly generalized) eigenvector q(x), corresponding eigenvalue E(x) and block Ω(x) satisfying the following conditions. Block dimensions and block extensions. For all x ∈ X
Ω(x) ⊂ {a : a ≤ λ} (2.3) #Ω(x) ≤ µ.
Perturbations of linear quasiperiodic system
Block overlapping. For all x ∈ X #
Ωa (x) ≤ µ.
15
(2.4)
Ωa (x)∩Ω(x)=0
If the blocks do not overlap, then the matrix D is a block matrix with blocks of dimension ≤ µ. In general the blocks do overlap but we shall require that resonant blocks don’t. Resonant block separation. For all x ∈ X  E a (x) − E(x) ≤ ρ =⇒ Ωa (x) = Ω(x) or dist(Ωa (x), Ω(x)) ≥ ν ≥ 1
(2.5)
Partition. There is a locally ﬁnite partition P such that D and Ω are smooth on the partition P.
(2.6)
By a matrix D being smooth on a partition we mean that each Dab is smooth on, a neighborhood of the closure of, the pieces of ∨{Tc (P) : c +
b−a b+a ≤ }. 2 2
By a set Ω being smooth on a partition we mean that each the characteristic function χΩb (a) is smooth on, a neighborhood of the closure of, the pieces of the same partition. This smoothness condition is consistent with the covariance. Remark. This concept of piecewise smoothness in relation to covariance is a little awkward. If A and B are two covariant matrices which are smooth on the partition, then AB will not be smooth on any partition unless A or B are truncated at some ﬁnite distance from the diagonal. If A or B are truncated at distance N , then AB is smooth on the reﬁned partition P(N ) =: ∨{Tc (P) : c ≤ N }. Matrices like eA or (I + A)−1 will in general not be be piecewise smooth with the deﬁnition we have chosen. Given a partition P we let Pδ denote another type of reﬁnement. Each piece Y of Pδ is contained in a piece of P and has “diameter” less than δ, i.e. any two points x, y in Y can be joined by a curve in Y of length less than δ – if Y is convex then δ is the usual diameter of Y . In the sequel we shall construct partitions like P(N1 )δ1 (N2 )δ2 (N3 ) . . . .
16
L. H. Eliasson
Deﬁnition. We say that a covariant matrix D is on normal form N F(α, β, γ, λ, µ, ν, ρ) if both D and D∗ satisfy (2.1 − 6) with the same block Ω and partition P. ¯D . The eigenvectors qD The eigenvalues will be complex conjugate, ED∗ = E and qD∗ of course not related unless D is symmetric, in which case they are equal, or Hermitian, in which case the are complex conjugate. We also use notations like N F(α, β, γ, λ, µ, ν, ρ; q, E, Ω, P) when we want to stress these objects. Remarks. 1. The parameters β, γ, λ, µ can be increased and the parameters α, ν, ρ can be decreased. For simplicity we shall assume that β, γ both are ≥ 1. 2. It follows from the deﬁnition that D is truncated at distance 2λ from the diagonal and that, for given a (or b), Dab = 0 for at most µ2 many b’s (or a’s). 3. Using the generalized Young inequality [27] we get an estimate of D in the operator norm  D C k ≤ β(
eα + 1 dim L k ) γ eα − 1
∀k ≥ 0.
(2.7)
If D is a m × mdimensional matrix then we also have  D C k ≤ mβγ k
∀k ≥ 0.
(2.8)
The estimate (2.8) is better than (2.7) if α is small but worse if m is large. We have the corresponding notion for multilevel operators on l2 (L) ⊗ CN . We then have N eigenvectors q j , each with corresponding eigenvalue E j and b,j block Ωj . (2.12) should hold for the matrix elements Da,i and (2.3) should hold for the blocks. (2.4) and (2.5) takes the form: for all j and for all x ∈ X
#
Ωa,i (x) ≤ µ;
(2.4 )
Ωa,i (x)∩Ωj (x)=0
for all i, j and for all x ∈ X  E a,i (x) − E j (x) ≤ ρ =⇒ Ωa,i (x) = Ωj (x) or dist(Ωa,i (x), Ωj (x)) ≥ ν ≥ 1.
(2.5 )
Perturbations of linear quasiperiodic system
3
17
Block splitting
In this section we shall see that we can conjugate a normal form matrix to a true block diagonal matrix. The price to pay is in the smoothness and will be related to almostmultiplicity of the eigenvalues. Proposition 1. Let D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P). Then there exists a subspace Λ(x) which is invariant under D(x) such that all eigenvalues of Λ∗ DΛ are ρclose, and which has the following properties: (i)
for all x Λ(x) ⊂ CΩ(x) ;
(3.1)
(ii)  Λ C k ≤ ((const
βµ2 2µ k ) γ) ρ
∀k ≥ 0;
(3.2)
(iii) Λ is smooth on the partition P(3λ)δ , where δ = (const
ρ 2µ 1 ; ) βµ2 γ
(3.3)
(iv) the angle between Λ(x) and the invariant subspace Λb (x), Λb (x) = τb Λ(Tb x), Λb (x)=Λ(x)
is ≥ (const
ρ µ2 ) . βµ2
(3.4)
The constants only depend on the dimensions of L and X. ˜ Proof. Let Ω(x) = ∪Ωa (x) where the union is taken over all a such that a Ω (x) ∩ Ω(x) = ∅. Then by (2.3 − 4), for each x,
˜ #Ω(x) ≤µ ˜ Ω(x) ⊂ {a ∈ L : a ≤ 3λ}. Consider DΩ˜ . This matrix is smooth on P(3λ) and by the remark (2.8) we have the estimate  DΩ˜ C k ≤ µβγ k ∀k ≥ 0.
18
L. H. Eliasson
Apply Lemma A.6 with ρ = r to get a DΩ˜ (x)invariant decomposition ˜
CΩ(x) =
k
˜ i (x) Λ
i=1
smooth on P(3λ)δ , with δ satisfying (3.3), and 2
˜ i C k ≤ ((const βµ )2µ γ)k Λ ρ
∀k ≥ 0.
˜ 1 (x) say. Let now x be ﬁxed. If q˜ is q(x) belongs to one of these spaces – Λ ˜ 1 (x) then it follows from Lemma A.7 an eigenvector of DΩ˜ (x) which lies in Λ that either q˜ is an eigenvector of D(x) or is perpendicular to CΩ(x) . Hence, if q˜a = 0 for some a ∈ Ω(x), then q˜ is an eigenvector of D(x) and hence = q b (x) for some b. This implies that  E(x) − E b (x) ≤ ρ and by (2.5) that Ωb (x) = Ω(x). Hence, either q˜ is supported in Ω(x), in which case it is an eigenvector of D(x), or it is supported in the complement of Ω(x). Therefore ˜ 1 (x) is an invariant space for D(x) which satisﬁes the same Λ(x) = CΩ(x) ∩ Λ ˜ 1 , i.e. (3.23). This proves (i), (ii) and (iii). estimates as Λ The angle between Λ and Λb follows also from Lemma A.6 since each ˜ Λb (x), which is not orthogonal to Λ(x), is contained in C Ω(x) and therefore ˜ i. is a subspace of some Λ When D is Hermitian in Proposition 1, then the angle between the invariant spaces is π2 ; the radius δ in (3.2) is const
ρ . βγµ2
In the multilevel case we get one subspace Λj (x) for each j = 1, . . . , N with j Λj (x) ⊂ CΩ (x) , and the matrix Q is deﬁned by Q(x) = (. . . Λb (x) . . . ), where Λb is the subspace (Λb,1 , . . . , Λb,N ). All the estimates are the same and the proof is also the same with obvious modiﬁcations.
4
Quadratic convergence
In this section we shall construct a conjugation which transforms a normal form plus a perturbation to a new normal form plus a smaller perturbation. The formulation will involve an auxiliary matrix W . The reason for this matrix is that we are not allowed to take inverses because they are not piecewise smooth, but we have to use approximate inverses. The role of W is to measure this approximation.
Perturbations of linear quasiperiodic system
19
Lemma 2. Let D ∈ N F(α, . . . , ρ; E, P) and let F be a covariant matrix, smooth on P and satisfying  Fab C k ≤ εe−αb−a γ k
∀k ≥ 0.
(4.1)
Then there exist covariant matrices K and G satisfying
[D, K] = F − G a b < qD if  E a (x) − E b (x) > ρ, ∗ (x), G(x)qD (x) >= 0 with the following properties: (i)
for all k ≥ 0
2 3µ3 6λα −αb−a k  Kab C k +  Gba C k ≤ ε( βµ1 2 )(const βµ e e (γ ) ρ ) 2
3µ γ = (const βµ ρ )
3
+4µ2
γ
(4.2)
(ii) K and G are smooth on the partition P(3λ)δ (6λ) where δ = (const
ρ 2µ 1 ; ) βµ2 γ
(4.3)
(iii) if F is truncated at distance λ from the diagonal, then K and G are truncated at distance λ + 6λ. The constants only depend on the dimensions of L and X. Proof. Let Λ be the invariant space of D deﬁned in Proposition 1, with ρ replaced by ρ3 , and let Q = (. . . Λa Λb . . . ). The estimate of Q follows from that of Λ, and Q is truncated at distance λ from the diagonal. Let ˜b . . . ) ˜ = (. . . Λ ˜ aΛ Q ˜∗, be the same matrix as Q but obtained from D∗ . The inverse Q−1 = B Q where B is a block matrix with the blocks ˜ a )∗ Λa ]−1 . [(Λ It follows that Q−1 is truncated at distance 2λ from the diagonal, is smooth on P(3λ)δ (λ) and satisﬁes  Q−1 C k ≤ (const – see Lemma A.6.
βµ2 2µ2 βµ2 4µ2 k ) ((const ) γ) ρ ρ
∀k ≥ 0
20
L. H. Eliasson
˜ = Q−1 DQ is a block diagonal matrix with blocks – which we Then D ˜ a – corresponding to the subspaces Λa , which is smooth on here denote by D a P(3λ)δ (3λ) and satisﬁes 2
2
˜ b C k ≤ βµ(const βµ )2µ2 e3λα e−αb−a ((const βµ )4µ2 γ)k D a ρ ρ
∀k ≥ 0.
˜ a and D ˜ b have two eigenvalues that ˜ a are ρ close, so if D Eigenvalues of D a a b 3 ˜ a and D ˜ b respectively diﬀer diﬀer by at least ρ then any two eigenvalues of D a b ρ by at least 3 . ˜ Let now F˜ = Q−1 F Q. Then F˜ will be smooth on the same partition as D and satisfy the same estimate but with the ﬁrst factor β replaced by ε. The equation becomes ˜b ˜b ˜b ˜b ˜ aK ˜b D a a − Ka Db = Fa − Ga . ˜ b respectively diﬀer by ≤ ρ, then the ˜ a and D If any two eigenvalues of D a b equation reduces to ˜ b = 0 and G ˜ b = F˜ b , K a a a and if at least two eigenvalues diﬀer by > ρ, then the equation becomes ˜ bD ˜b ˜ ab − K ˜b ˜ aa K D a b = Fa
˜ b = 0. and G a
˜ is clear so we only need to estimate K ˜ from the second The estimate of G ˜ a by a unitary transformation we equation. Since we can triangularize each D a get 2 ˜ b C 0 ≤ ε[ βµ (const βµ )2µ2 ]µ e3λα e−αb−a . K a ρ ρ Since the solution of the second equation is unique we get estimate of the derivatives by diﬀerentiating the equation. ˜ −1 and G = QGQ ˜ −1 . They will satisfy the required We now let K = QKQ estimates and be smooth on P(3λ)δ (6λ). (iii) is clear by construction. Proposition 3. Let D ∈ N F(α, . . . , ρ; Ω, P) and let F and W be covariant matrices, smooth on P and satisfying  Fab C k +  Wab C k ≤ εe−αb−a γ k
∀k ≥ 0.
Then there exists a constant C – depending only on the dimensions of X and L – such that if ε ≤ ξ(
ρ − ρ (α − α )dim L )µ , β
then there exist
ξ =: (C
ρ 3µ3 −8λα ) e , βµ2
D ∈ N F(α , β , γ , λ, µ, ν, ρ ; Ω, P )
(4.4)
Perturbations of linear quasiperiodic system
21
and covariant matrices U , V , F and W such that V (D + F )U = D + F
and
V (I + W )U = I + W
with the following properties: (i)
α < α, 2
3µ γ = ( C1 βµ ρ )
3
+4µ2
β = (1 + ξε )β,
γ,
ρ < ρ;
(4.5)
(ii) for all k ≥ 0,  (U − I)ba C k +  (V − I)ba C k +  (D − D)ba C k ≤
ε −αb−a k e (γ ) , ξ (4.6)
and D − D is truncated at distance ν from the diagonal; (iii) for all k ≥ 0,
 (F )ba C k +  (W )ba C k ≤ ε e−α b−a (γ )k 2 ε −(ν−8λ)(α−α ) ε = max[ ξ2 (α−α ]; )dim L , εe
(4.7)
(iv) F and W are smooth on P = P(5λ)δ (6λ + 2ν) where δ = (C
ρ 2µ 1 . ) βµ2 γ
(4.8)
Notice that the smallness assumption only depends on α, β, λ, µ, ρ, α − α , ρ − ρ and that the size of the new perturbation F depends on ν. Nothing depends on γ. ˆ be the truncations of F and W at distance ν − 8λ from Proof. Let Fˆ and W the diagonal. Deﬁne K and G so that 1 ˆ ˆ ). D + DW [K, D] + G = Fˆ − (W 2 (This choice gives an Hermitian G and an antiHermitian K whenever D, F, W are Hermitian.) The right hand side is smooth on P(2λ), truncated at distance ν − 6λ from the diagonal and satisﬁes  (RHS)ba C k ≤ constεβµ2 e2λα e−αb−a γ k
k ≥ 0,
because D is truncated at distance 2λ and Dcd = 0 for at most µ2 many c’s or d’s.
22
L. H. Eliasson
Hence, by Lemma 2 we have  Kab C k +  Gba C k ≤
ε −αb−a k e (γ ) ξ
and K and G are smooth on the partition P(5λ)δ (6λ). Moreover, K and G are both truncated at distance ν. If we deﬁne 1 ˆ , U =I +K − W 2
1 ˆ V =I −K − W 2
then U, V and G will satisfy the estimate (4.6) and ˆ ) + O2 (W, W ˆ , K)] + [O3 (W, W ˆ , K)] V (I + W )U − I = [(W − W =: W = W2 + W3 and ˆ , K)] + [O3 (F, W ˆ , K)] V (D + F )U − D − G = [(F − Fˆ ) + O2 (F, W =: F = F2 + F3 . Estimating this using Lemma A.8 gives that Fj and Wj satisﬁes (4.7). Observing that W and F are at most linear in the nontruncated matrices W, F we see that they are smooth on the partition P(5λ)δ (6λ + 2ν). It is clear that D = D + G satisﬁes (2.14) with α , β , γ , λ, µ and (2.6) with P Since the eigenvalues of DΩ and DΩ diﬀer by at most 4(const
β αdim L
1
)1− µ (const
ε ξαdim L
1
)µ ≤
1 (ρ − ρ ) 4
(Lemma A.2), where we used (2.7) to estimate the norm of the matrices, it follows that  (E )a (x) − (E )(x) ≤ ρ → E a (x) − E(x) ≤ ρ. Therefore D veriﬁes (2.5).
We now iterate the construction in Proposition 3 a ﬁnite number of times in order to improve the result. Corollary 4. Let D ∈ N F(α, . . . , ρ; Ω, P) and let F and W be covariant matrices, smooth on P and satisfying  Fab C k +  Wab C k ≤ εe−αb−a γ k
∀k ≥ 0.
Perturbations of linear quasiperiodic system
23
Then there exists a constant C – depending only on the dimensions of X and L – such that if
dim L µ ) ] ε ≤ min[ξ 2 (α − α )dim L , ξ( ρ−ρ β (α − α ) (4.9) ρ 3µ3 −8λα , ξ =: (C βµ2 ) e then, for any integer 1 1 [log((ν − 8λ)(α − α )) − log log( )], 2 log 2 ε
1 ≤ n ≤ there exist
D ∈ N F(α , β , γ , λ, µ, ν, ρ ; Ω, P )
and covariant matrices U , V , F and W such that V (D + F )U = D + F
and
V (I + W )U = I + W
with the following properties: (i)
α < α,
γ =
2 3µ3 +4µ2 ( C1 βµ γ, ρ )
β = (1 + ξε )β
ρ < ρ;
(4.10)
(ii) for all k ≥ 0,  (U − I)ba C k +  (V − I)ba C k +  (D − D)ba C k ≤
ε − α b−a k e 2 (γ ) , ξ (4.11)
and D − D is truncated at distance ν from the diagonal; (iii) for all k ≥ 0,
 (F )ba C k +  (W )ba C k ≤ ε e−α b−a (γ )k n ε = ( ξ2 (α−αε )dim L )2 ; (iv) F and W are smooth on P = Rn (P), where
R(P) = P(5λ)δ (6λ + 2ν) ρ 2µ 1 δ = (C βµ 2) γ.
(4.12)
(4.13)
Proof. We can assume that 2α > α. Let α = α1 > α2 > . . . and ρ = ρ1 > ρ2 > . . . with αi − αi+1 = 2−i (α − α ),
ρi − ρi+1 = 2−i (ρ − ρ ),
24
L. H. Eliasson
and let εi+1 = max[
ε2i , εi e−(ν−8λ)(αi −αi+1 ) ] ξ 2 (αi − αi+1 )dim L ε2i = 2 ξ (αi − αi+1 )dim L
i ≤ n.
If we apply n times Proposition 3 we get the statement (iiv), but only with 1 βµ2 (3µ3 +4µ2 )n ) γ. γ = ( C ρ In order to get better smoothness we must look more closely. Let us deﬁne
Vj . . . V1 (D + F )U1 . . . Uj = Dj+1 + Fj+1 Vj . . . V1 (I + W )U1 . . . Uj = I + Wj+1 . with 1 ˆ Uj = I + Kj − W j, 2
1 ˆ Vj = I − K j − W j 2
ˆj Dj+1 = Dj + G
and
1 ˆ ˆ [Kj , Dj ] + (Dj+1 − Dj ) = Fˆj − (W j Dj + Dj Wj ), 2 where ˆ denotes truncation at distance ν − 8λ from the diagonal and D1 = D and F1 = F . We now conjugate these equations to the left and right by Q−1 and Q. This ˜ j and give the equations conjugation will block diagonalize all the Dj ’s to D ˜j, D ˜j] + G ˜ j = Q−1 Fˆj Q − 1 (Q−1 W ˜j + D ˜ j Q−1 W ˜ ˆ j QD ˆ j Q) = RHS. [K 2 These equations now split into blocks: ˜ j )ba = 0 (K
and
b ˜ j )b = (RHS) ˜ (G a a
˜ a and D ˜ b diﬀer by ≤ ρ; when any two eigenvalues of the blocks D a b b ˜ j )b (D ˜ j )b − (D ˜ j )a (K ˜ j )b = (RHS) ˜ (K a b a a a
and
˜ j )b = 0, (G a
˜ b diﬀer by ≥ ρ. That ˜ a and D when at least two eigenvalues of the blocks D a b the Kj ’s, Dj ’s and Fj ’s are γ smooth now follows by an easy induction. When D, F and W are Hermitian these results can be improved. In Lemma 2, G is Hermitian and K is antiHermitian – this is an immediate consequence of the construction we have made; the estimate in (4.2) becomes
4λα −αb−a k  Kab C k +  Gba C k ≤ constε βµ e (γ ) ρ e 2
2µ+1 γ = (const βµ γ; ρ )
Perturbations of linear quasiperiodic system
25
ρ the estimate for δ in (4.3) becomes (const βγµ 2 ). In Proposition 3 and Corollary 4 D , F and W are Hermitian and V = U ∗ ; the smallness assumptions (4.4) and (4.9) are given by
ε ≤ ξ(
ρ − ρ (α − α )dim L ), β
ξ = (C
and ε ≤ min[ξ 2 (α − α )dim L , ξ(
ρ )e−6λα βµ2
ρ − ρ (α − α )dim L )] β
respectively; the estimate for γ in (4.5) and (4.10) becomes γ = (
βµ2 2µ+1 ) γ; Cρ
ρ the estimate of δ in (4.8) and (4.12) becomes (C βγµ 2 ). In the multilevel case the proof goes through in the same way and all the results remain the same with obvious modiﬁcations.
5
Block clustering
The result in Corollary 4 depends on ν – it is better the larger ν is. In general we cannot increase ν, but it may happen that the Ωblocks of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω) cluster into bigger blocks with a better separation property. Indeed it may happen that there are unions of blocks Ω = ∪i Ωai – the union depends on x – such that  E b (x) − E c (x)  ≤ ρ =⇒ (Ω )b (x) = (Ω )c (x) or dist((Ω )b (x), (Ω )c (x)) ≥ ν .
(5.1)
Of course Ω (x) is likely to be bigger so Ω (x) ⊂ {a : a ≤ λ }
and
#Ω ≤ µ .
(5.2)
Since D is on normal form and therefore satisﬁes (2.5) it is natural to expect that dist(Ωai (x), Ωaj (x)) ≥ ν
(5.3)
for all smaller blocks Ωai building up Ω . Deﬁnition. The Ωblocks of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω) are said to be C(λ , µ , ν ) − clustering into Ω − blocks if (5.13) hold.
26
L. H. Eliasson
Then we get a better result. Proposition 5. Let D ∈ N F(α, . . . , ρ; Ω, P) be such that the Ωblocks are C(λ , µ , ν )clustering into Ω blocks. Let F and W be covariant matrices, both smooth on P and satisfying  Fab C k +  Wab C k ≤ εe−αb−a γ k
k ≥ 0.
Then there exists a constant C – depending only on the dimensions of X and L – such that if
dim L µ ε ≤ min[ξ 4 (α − α )2 dim L , ξ( ρ−ρ ) ], β (α − α ) (5.4) ρ 3µ3 −8λα , ξ =: (C βµ2 ) e then there exist D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P ), truncated at distance ν from the diagonal, and covariant matrices U , V , F and W such that V (D + F )U = D + F
and
V (I + W )U = I + W
with the following properties: (i)
α < α,
γ =
β = (1 +
2 7µ3 ( C1 βµ γ, ρ )
√
ε)β
ρ < ρ;
(ii) for all k ≥ 0  (U − I)ba C k +  (V − I)ba C k +  (D − D)ba C k ≤
(5.5)
√ − α b−a k εe 2 (γ ) (5.6)
and D − D is truncated at distance ν from the diagonal; (iii) for all k ≥ 0,
 (F )ba C k +  (W )ba C k ≤ ε e−α b−a (γ )k √ 1 ε = e− 2 (ν −8λ)(α−α ) ;
(5.7)
(iv) F and W are smooth on P = Rn (P), where
R(P) = P(5λ)δ (6λ + 2ν ),
1 1 n = 2 log 2 [log((ν − 8λ)(α − α )) − log log( ε )] ρ 2µ 1 δ = (C βµ2 ) γ .
(5.8)
Perturbations of linear quasiperiodic system
27
Proof. The assumption implies that D ∈ N F(α, β, γ, λ , µ , ν , ρ; Ω , P). Applying n times Corollary 4 would give the result, but with a smallness condition (5.4) that depends on λ , µ . But the occurrence of the block dimensions and block extensions enters through the block splitting and this only depends on λ, µ since D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P). The block dimension µ enters however in the estimate of  E −E  which is reﬂected in the second smallness condition in (5.4). Though Proposition 5 is not a corollary of Corollary 4, the proof is word by word the same as for Corollary 4. 5.1
Choice of parameters
How good must this clustering property be in order for us to apply Proposition 5 iteratively? We want sequences αj , βj , γj , λj , µj , νj , ρj , εj satisfying for all j ≥ 1 √ βj+1 = (1 + 2εj )β3j βj µ (5.9) γj+1 = ( C1 ρj j )7µj γj √ ε − 12 (νj+1 −8λj )(αj −αj+1 ) =e j+1
εj ≤ ε ≤ j
3
min[(C βj µj 2 )12µj e−32λj αj (αj − αj+1 )2 dim L , ρ
j
ρj −ρj+1 (αj − βj −2j 2 dim L e αj ,
(
αj+1 )dim L )2µj+1 ]
(5.10)
where C is the constant of Proposition 5. If we assume that εj ≤ e−j , then βj ≤ constβ1 for all j, and it suﬃces that εj ≤ min[(
Cρj 36µ3j −96λj αj ) ,e , (αj − αj+1 )6 dim Lµj+1 , β1 µ2j ρj − ρj+1 4µj+1 −4j ) , e ]. ( 2β1
How shall we choose the sequences αj , λj , µj , νj , ρj so that (5.10) is fulﬁlled for all j ≥ 1. In order to simplify the discussion we assume a weak decay condition on the αj ’s and the ρj ’s: αj − αj+1 ≥ 4−j αj
and ρj − ρj+1 ≥ 4−j ρj
∀j ≥ 1.
These requirements leads to the conditions √ √ νj+1 αj νj+1 αj 4j − 1 ) ≤ αj+1 ≤ min[ j αj , j+6 exp(− j+6 ] 2 µj+2 dim L 4 2 λj+1
(5.11)
28
L. H. Eliasson
and
√
νj+1 αj ρj+1 4j − 1 ρj ) ≤ ≤ β1 4j β1 2j+6 µj+2
(5.12)
log( α1j ) log(λj+1 )µj+2 log( β1 )µ j+2 ≥ 2j+6 dim L ρj λj αj 2β1 µ2 log( Cj+1 )µ3j+1 .
(5.13)
exp(− and √
νj+1 αj
where the ﬁrst two conditions make (5.1112) possible. Hence, if the C(λj+1 , µj+1 , νj+1 )clustering satisﬁes (5.13), then we can choose αj+1 and ρj+1 according to (5.1112), and (5.10) will hold for all j ≥ 1 if it holds for j = 1. Notice that in the Hermitian case µj+2 can be replaced by µj+1 in (5.1113). Proposition 6. Let D1 ∈ N F(α1 , . . . , ρ1 ; E1 , Ω1 , P1 ) and let W1 and F1 be covariant matrices, smooth on the P1 , with W1 = 0 and  (F1 )ba C k ≤ ε1 e−α1 b−a γ1k
∀k ≥ 0.
Then there exists a constant C – depending only on the dimensions of X and L – such that if the sequences αj , βj , γj , λj , µj , νj , ρj , εj satisfy (5.910) then the following hold. (I) If the Ω1 blocks are C(λ2 , µ2 , ν2 )clustering into Ω2 blocks, then there is an J ≥ 2 such that the statement (SJ ) holds: for all 1 ≤ j < J there exist Dj+1 ∈ N F(αj+1 , . . . , ρj+1 ; Ej+1 , Ωj+1 , Pj+1 ), truncated at distance νj+1 from the diagonal, and matrices Uj , Vj , Vj (Dj + Fj )Uj = Dj+1 + Fj+1 ,
Vj (I + Wj )Uj = I + Wj+1
that satisfy (5.6 − 8)j . (II) If the ΩJ blocks are C(λJ+1 , µJ+1 , νJ+1 )clustering into ΩJ+1 blocks, then also (SJ+1 ) holds. (III) If (S∞ ) holds, then there exists a matrix U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),
∀x ∈ X,
and D∞ (x) is a norm limit of Dj (x). If D and F are Hermitian, then U is unitary. (IV) Let ηj be the measure of the set of all x ∈ X such that
If
#{ a ≤ νj+1 + 2λj : Eja (x) − Ej (x) ≤ ρj } > 1. ηj < ∞, then D∞ (x) is pure point for a.e. x.
Perturbations of linear quasiperiodic system
29
Proof. (IIII) are direct consequences of Proposition 5. We get U and U −1 as the norm limit of the compositions U1 U2 . . . Uj If
and Vj . . . V2 V1 .
ηj < ∞, it follows that for a.e x
#{ a ≤ νj+1 + 2λj : Eja (x) − Ej (x) ≤ ρj } = 1,
∀j ≥ j(x).
This implies that the blocks stop increasing, i.e. Ωj (x) = Ωj(x) (x), for a.e. x ∈ X. This proves (IV).
∀j ≥ j(x),
In the multilevel case the proofs goes through in the same way and all the results remain the same with obvious modiﬁcations. We can describe this perturbative process with a diagram. ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 ) ⇓ ⇓
⇐⇐⇐
C(λ2 , µ2 , ν2 ) − clustering
⇓ ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 ) ⇓ ⇓
⇐⇐⇐
C(λ3 , µ3 , ν3 ) − clustering
⇓ ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 ) ⇓ ⇓
⇐⇐⇐
C(λ4 , µ4 , ν4 ) − clustering
⇓ Figure 1
6
Transversality of resultants
In this section we shall derive some estimates of the resultants of the matrices Dj constructed in Proposition 6, which will prove essential in showing the clustering property of blocks.
30
L. H. Eliasson
The resultant Res(D1 , D2 ) of two square matrices D1 and D2 of dimension m1 and m2 , respectively, is the product of the diﬀerences of the eigenvalues, i.e. (ei − fj ), where ei and fj are the eigenvalues of D1 and D2 respectively. All the eigenvalues are counted, so this is always a product of m1 m2 many factors. The resultant measures the diﬀerence between the spectra of the two matrices. For example  e1 − f1  ≤ ε =⇒ Res(D1 , D2 )  ≤ ε( D1 )  +  D2 )m1 m2 −1 . The resultant Res(D1 , D2 ) = det χD2 (D1 ), where χD2 (t) = det(D2 − tI) is the characteristic polynomial of the matrix D2 . χD2 (t) is a sum of ≤ m2 ! many monomials of degree m2 in the matrix elements of D2 − tI. Introducing D1 for t we get an m1 × m1 matrix whose elements are ≤ m2 !(m1 )m2 −1 many monomials of degree m2 in the matrix elements of these two matrices and in their diﬀerences. Taking the determinant gives this number to the power m1 , times m1 !. This shows that Res(D1 , D2 ) is a sum of less than (m1 m2 )m1 m2 many monomials of degree less than m1 m2 in the components of D1 and D2 and their diﬀerences. Moreover Res(D1 , D2 ) = det(D1 − fj I). ˆ 2 , then Res(D1 , D2 ) vanishes to the This formula shows that if D1 = D2 + εD order m1 = m2 in the variable ε. For a given normal form matrix D ∈ N F(α, . . . , ρ; Ω, P) consider, for any a, b ∈ L, the resolvents ua,b (x, y) = Res(DΩa (x+y) (x + y), DΩb (x) (x)). Since D is smooth on P we get immediately that ua,b is smooth when x, x + y vary over the pieces of the partition Ta−1 (P(λ)) ∨ Tb−1 (P(λ)) and that 2
 ua,b C k ≤ (4dim X µ2 β)µ γ k
∀k ≥ 0.
(6.1)
Since ua,a (x, y) vanishes to order #Ωa in y we get for ua,a (x, y) as a function of x, with y ﬁxed, (k + µ)! 2 k ) γ ∀k ≥ 0. (6.2) k! Consider now the conditions: for all x, y ∈ X and for all i a
2
 ua,a ( , y) C k ≤  y #Ω γ µ (4dim X µ2 β)µ (
1 ∂ k ua,b (x, y)  ≥ σ, (k!)2 γ k yi
(6.3)
a b 1 ∂xki ua,b (x, y)  ≥ σ  y #(Ω (x+y)∩Ω (x)) , 2 k (k!) γ
(6.4)
max 
0≤k≤J
max 
0≤k≤J
where J = #Ωa (x + y) × #Ωb (x)s and s is a ﬁxed parameter.
Perturbations of linear quasiperiodic system
31
Deﬁnition. We say that D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω) is (σ, s)transversal on the Ωblocks, denoted D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω)&T (σ, s), if the functions ua,b satisfy (6.34) for all x, y such that Ωa (x + y) and Ωb (x) are disjoint or equal. Lemma 7. Assume that D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P)&T (σ, s) is truncated at distance ν from the diagonal, and assume also that the Ωblocks are C(λ , µ , ν )clustering into Ω blocks. Let D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P ) be a perturbation of D  (D − D)ba C k ≤
√
εe− 2 b−a (γ )k α
∀k ≥ 0.
If 4
ε ≤ C (µ ) ( then
γ 2s(µ )2 σ 4s(µ )4 ) ( ) γ β
(6.5)
D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P )&T (σ , s)
for any 4
σ ≤ C (µ ) (
γ s(µ )2 σ s(µ )4 ) ( ) . γ β
(6.6)
The constant C depends only on the dimensions of L and X, and on s. Proof. Consider ﬁrst the case (6.3). Let u ˜c,d (x, y) = Res(D(Ω )c (x+y) (x + y), D(Ω )d (x) (x)) where
Ω c = ∪Ωa ,
(Ω )d = ∪Ωb ,
are unions of less than µ many Ωa ’s which are separated by a distance at least ν. Since D is truncated at distance ν from the diagonal we get that u ˜c,d (x, y) = ua,b . a,b
Then by Lemma B2 max
0≤k≤s(µ
)2

1 (k!)2 (γ )k
2 σ 4 1 2 4 γ ∂yki u ˜c,d (x, y)  ≥ ( )8s (µ ) ( )s(µ ) ( )s(µ ) . (6.3) 2 γ β
32
L. H. Eliasson
Let now uc,d (x, y) = Res(D(Ω )c (x+y) (x + y), D(Ω )d (x) (x)).
Then uc,d (x, ) C k ≤  uc,d (x, )−˜
√
2
ε(4dim X (µ )2 )(µ )
+1
√ 2 (β+ ε)(µ ) (γ )k
∀k ≥ 0
which gives the result. The case (6.4), when the blocks are disjoint, is proven in the same way. In order to see (6.4) when the blocks are equal we consider va,b (x, y) = ua,b (x, y) when Ωa (x + y) ∩ Ωb (x) = ∅ and va,b (x, y) = ua,b (x, y)
1 y
#Ωa (x+y)
,
when Ωa (x + y) = Ωb (x). We deﬁne v˜c,d and vc,d in the same way as u ˜c,d and uc,d using v instead of u. We now apply the same argument to the v:s as before to the u’s. 6.1
Choice of parameters
We want sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfying for all j ≥ 1 √ βj+1 = (1 + εj )βj 2 γj+1 = ( 1 βj µj )7µ3j γj C √ ρj (6.7) εj+1 = e− νj+1 −8λj )(αj −αj+1 ) 4 3 2 4 σj+1 = C µj+1 (C ρj 2 )14sµj µj+1 ( σj )sµj+1 , βj βj µ j
and, besides (6.8)=(5.10), 4
εj ≤ C µj+1 (
γj 4sµ2j+1 σj 2sµ4j+1 ) ( ) , γj+1 βj
(6.9)
where C is the smallest of the constant in Proposition 5 and Lemma 7. How shall we choose the sequences λj , µj , νj , αj , ρj so that the two conditions (6.89) are fulﬁlled? If we assume that µj+1 ≥ 4sµj it follows that σj+1 ≤ (
∀j ≥ 1
Cσ1 ρj (µ1 ...µj+1 )5 ) . β1
Then we are lead to (6.10)=(5.11), √ νj+1 αj ρj+1 4j − 1 ρj ) ≤ ≤ exp(− j+6 2 (µ1 . . . µj+2 )5 β1 4j β1
(6.11)
Perturbations of linear quasiperiodic system
and
√
νj+1 αj
log( α1j ) log(λj+1 )µj+2 log( 1 )(µ1 . . . µj+2 )5 ≥ (2s)j+6 (dim L) ρj λj αj β1 µ2 log( Cσj+1 )(µ1 . . . µj+2 )5 . 1
33
(6.12)
The clustering required by (6.12) is much stronger than (5.13) but it will be fulﬁlled in the applications we have in mind. Proposition 8. Let D1 ∈ N F(α1 , . . . , ρ1 ; E1 , Ω1 , P1 )&T (σ1 , s) and assume that the D1 is truncated at distance ν1 from the diagonal. Let W1 and F1 be covariant matrices, smooth on P1 with W1 = 0 and  (F1 )ba C k ≤ εγ1k
∀k ≥ 0.
Then there exists a constant C – depending only on the dimensions of X and L and on s – such that if the sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfy (6.79) then the following hold. (I) If the Ω1 blocks are C(λ2 , µ2 , ν2 )clustering into Ω2 blocks, then there is an J ≥ 2 such that the statement (SJ ) holds: for all 1 ≤ j < J there exist Dj+1 ∈ N F(αj+1 , . . . , ρj+1 ; Ej+1 , Ωj+1 , Pj+1 )&T (σj+1 , s) and matrices Uj , Vj , Vj (Dj + Fj )Uj = Dj+1 + Fj+1 ,
Vj (I + Wj )Uj = I + Wj+1
that satisfy (5.6 − 8)j . (II) If the ΩJ blocks are C(λJ+1 , µJ+1 , νJ+1 )clustering into ΩJ+1 blocks, then also (SJ+1 ) holds. (III) If (S∞ ) holds, then there exists a transformation U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),
∀x ∈ X,
and D∞ (x) is a norm limit of Dj (x). If D and F are Hermitian, then U is unitary. (IV) Let ηj be the measure of all x ∈ X such that
If
#{ a ≤ νj+1 + 2λj : Eja (x) − Ej (x) ≤ ρj } > 1. ηj < ∞, then D∞ (x) is pure point for a.e. x.
Proof. Immediate consequence of Proposition 6 and Lemma 7.
In the multilevel case the transversality condition applies to the resultants u(a,i),(b,j) (x, y) = Res(DΩa,i (x+y) (x + y), DΩb,j (x) (x)).
34
L. H. Eliasson
The proofs goes through in the same way and all the results remain the same with obvious modiﬁcations. We can describe this perturbative process with the diagram in Figure 2. In the next section we shall consider only matrices over T with a Diophantine quasiperiodic group action, and then we shall see that transversality on blocks implies clustering of blocks and the diagram will then be closed. ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 )
&
⇓ ⇓
⇓ ⇐⇐⇐
C(λ2 , µ2 , ν2 ) − clustering
⇓ ⇒⇒⇒ &
⇓
T (σ2 , s) ⇓
⇐⇐⇐
C(λ3 , µ3 , ν3 ) − clustering
⇓
⇓ ⇓
ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 )
⇒⇒⇒ &
⇓ ⇓
⇓ ⇓
ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 ) ⇓
T (σ1 , s)
T (σ3 , s) ⇓
⇐⇐⇐
C(λ4 , µ4 , ν4 ) − clustering
⇓
⇓ ⇓
Figure 2
7
A Perturbation theorem
In this section we shall restrict the discussion to the circle X = T and the Laction
T :L×T→T (7.1) (a, x) → (x+ < a, ω >) where ω is a Diophantine vector < a, ω > ≥
κ  a τ
∀a ∈ L \ {0}
(7.2)
– τ > dim L and κ > 0 are ﬁxed numbers. The numbers τ, κ will appear in the estimates but since they will be ﬁxed throughout the iteration we shall not make them explicit in our notations. We shall use the Diophantine property in Lemma 9 and also in Lemma 11 where we shall make a particular choice of the partitions Pj . In Lemma 10 we make use of the ﬁrst transversality condition (6.3), and in the proof of Theorem 12 we also use the second transversality condition (6.4).
Perturbations of linear quasiperiodic system
35
Deﬁnition. We say that the eigenvalues of D ∈ N F(α, . . . , ρ; E)&T (σ, s) have almostmultiplicity ≤ µµ if, for all x and for all t ≤ ρ, the inequality  E(x + y) − E(x) > t( is fulﬁlled outside at most
µ µ
µ ) µ
many intervals of length less than 1 t 1 2(4βµ ) s ( ) sµ2 . σ
Lemma 9. Let the eigenvalues of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&T (σ, s) have almostmultiplicity ≤
µ µ
and assume
ρ ≤ ( Then
κ 1 2τ sµ2 ) σ. 4βµ λ
(7.3)
D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&C(λ , µ , ν )
for any ν =
1 µ κ 1 σ τ sµ ( )τ ( ) 2 , 2µ 4βµ ρ
λ =
Proof. For each x,  E(x + y) − E(x) > ρ( is fulﬁlled outside at most
µ µ
µ (ν + 2λ). µ
(7.4)
µ ) µ
many intervals of length less than
1 1 ρ L = 2(4βµ ) s ( ) sµ2 . σ
If < a, ω >≤ L it follows that κ 1  a  ≥ ( ) τ = S. L This means that on distances of size less than S there are at most µ µ
µ µ
many
many resonant blocks. ρalmost resonant eigenvalues and, hence, at most Each resonant block extends over 2λ sites so there must be a gap of size at µ least µµ (S − 2λ µµ ) ≥ 2µ between the resonant blocks. And unless S = ν there is a gap of size ν between the resonant blocks, they will extend over distances which are at most λ = µµ (ν + 2λ). This proves the lemma.
36
L. H. Eliasson
We must now take into account the partition P of T. We shall for this discussion ﬁx an integer p and deﬁne  P  to be the minimum of the length of the pieces, with the exception of the p − 1 smallest pieces. Hence, all pieces of P with the exception of at most p − 1 pieces have a length larger than this number. Lemma 10. Let D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&T (σ, s). (i) Then the eigenvalues of D have almostmultiplicity ≤
µ µ
with
2 2 γ µ = 2sµ 16 (4βµ2 )µ (sµ2 )2 (p+  P(λ) −1 ). µ σ
(ii) Assume that the Ωblocks are C(λ , µ , ν )clustering into Ω blocks, and let
D ∈ N F(α , β , γ , λ , µ , ν , ρ ; E , Ω , P )&T (σ , s)
be a perturbation of D,  (D − D )ba C 0 ≤
√
εe−α b−a .
If ε ≤ min[(
2 1 8sµ2 (µ )3 σ 4sµ2 µ ) ( ) ,  P (λ ) 4sµ µ ] 2sβ µ γ
(7.5)
and 1
ρ ≤ ε µ ,
(7.6)
then the eigenvalues of D have almostmultiplicity ≤
µ µ
with
2
µ = p22s(µ ) .
(7.7)
Proof. Let u(x, y) be the resultant of DΩ(x+y) (x + y) and DΩ(x) (x). u(x, y) is smooth for y ∈ I, x + I ∈ P(λ), and satisﬁes ˜ k = (4βµ2 )µ2 γ k  u(x, ) C k ≤ β(γ)
∀k ≥ 0.
The eigenvalues of D are bounded by βµ, so if  E(x + y) − E(x)  ≤ t( it follows that  u(x, y)  ≤ t(
µ ), µ
2 µ )(2βµ)µ . µ
t ≤ ρ,
Perturbations of linear quasiperiodic system
37
We can now apply Lemma B1 to each interval I. It follows that, for each x and each t ≤ ρ µ  E(x + y) − E(x) > t( ) µ µ µ
is fulﬁlled outside at most
many intervals of length less than
1 t 1 Lt = 2(4βµ ) s ( ) sµ2 , σ
i.e. the eigenvalues of D have almostmultiplicity ≤ µµ . In order to prove (ii) notice that the eigenvalues of DΩ and DΩ diﬀer by 1 at most 2β µ ε 2µ . Using (i) it follows that  E (x + y) − E (x)  > is fulﬁlled outside at most
µ µ
t µ ( ) 2 µ
many intervals of length less than Lt if we let 1
t = 4β µε 2µ . By the second part of (7.5), each such interval is cut by the partition P (λ ) into at most p subintervals I. By (7.6), ρ ( µµ ) ≤ 2t ( µµ ) so we can restrict to one of these intervals I. Let u (x, y) be the resultant of DΩ (x+y) (x + y) and DΩ (x) (x). u (x, y) is smooth for y ∈ I and satisﬁes 2  u (x, ) C k ≤ β˜ (γ )k = (4β (µ )2 )(µ ) (γ )k
∀k ≥ 0.
The eigenvalues of D are bounded by β µ , so if  E (x + y) − E (x)  ≤ t (
µ ), µ
t ≤ ρ ,
it follows that
2 µ )(2β µ )(µ ) . µ By Lemma B1 this deﬁnes a union of at most
 u (x, y)  ≤ t (
γ ˜ 8β (s(µ )2 + 1)2  I  +1) σ many intervals, each of which has length at most 2
2s(µ ) (
1 1 t 2(4β µ ) s ( ) s(µ )2 . σ By the ﬁrst condition in (7.5) the number of intervals is less than
1 2s(µ )2 2 µ
38
L. H. Eliasson
7.1
Choice of parameters
We want sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfying for all j ≥ 1 √ βj+1 = (1 + εj )βj 2 1 βj µj 7µ3j γ = ( γj j+1 C ρj√ ) 1 − (ν −8λj )(αj −αj+1 ) j+1 2 εj+1 = e 4 3 2 4 ρ σ σj+1 = C µj+1 (C βj µj 2 )14sµj µj+1 ( βjj )sµj+1 j µ λj+1 = ( µj+1 )(νj+1 + 2λj ) j 2 2sµ2j µj+1 = p˜2 , p˜ = 16 σγ11 (4β1 µ21 )µ1 (sµ21 )2 (p+  P1 (λ1 ) −1 ) 1 2 1 σ µj νj+1 = 2µj+1 ( 8βj µκj+1 ) τ ( ρjj ) 4τ sµj ,
(7.8)
where C is the smallest of the constants in Proposition 5 and Lemma 7. The question is now to choose αj and ρj so that, besides (7.9)=(6.8) and (7.10)=(6.9), also 3 3 εj ≤ min[( 2sβ 1 µ σγj+1 )4sµj+1 ,  Pj+1 (λj+1 ) 4sµj+1 ] j+1 j+1 j+1 1 (7.11) 1 2τ sµ2j ρj ≤ min[ε µj , ( κ σj ]. j−1 8µj+1 βj λj ) Now λj increases at least as fast as 2j , so if we deﬁne αj =
1 λj
(7.12) 1 µ
j , ρj will also decay then αj will decay exponentially fast. Since ρj ≤ εj−1 fast. Then the conditions are fulﬁlled if ρj satisfy 6 εj ≤ min[( Cσ1 )(µ1 ...µj+1 )6 , ρ(µ1 ...µj+1 ) ,  Pj+1 (λj+1 ) 4sµ3j+1 ] j β1 1 ρ ≤ ε µj .
j
j−1
If we let (µ1 ...µj+1 )6
ρj
= εj ,
(7.13)
which deﬁnes ρj inductively, then these conditions amounts to 1
ρj ≤ min[(
4sµ3
j+1 Cσ1 (µ1 ...µj )6 µj 6 ), ρj−1 ,  Pj+1 (λj+1 )  (µ1 ...µj+1 ) ]. β1
(7.14)
The ﬁrst two conditions are easily seen to be fulﬁlled because ρj decays superexponentially. We now must discuss the partition and Pj+1 (λj+1 ).
Perturbations of linear quasiperiodic system
39
Lemma 11. We can choose the partition P in Proposition 5 in such a way that the endpoints of the pieces of P (N ), for any N are contained in ∪{Ta (endpoints of P) : a ≤ (11λ + 2ν + where δ = (C
1 1 τ +1 ( ) )n + N }, κ δ
ρ 2µ 1 ) βµ2 γ
and
1 1 (log((ν − 8λ)(α − α )) − log log( )). 2 log 2 ε (When P has only one piece we can let an arbitrary point be its “endpoint”.) n =
Proof. Due to the Diophantine condition (7.2) any interval of length δ will contain at least one element of the orbit {Ta x : a ≤ κ1 ( 1δ )τ +1 } for any x, in particular for x ∈ P. Hence we can choose such points as endpoints of the partition into pieces of length δ. Since P = Rn P,
RP = P(5λ)δ (6λ + 2ν ),
the result follows. It follows that the partitions from Proposition 8 can be chosen so that the endpoints of Pj (λj ) are contained in ∪{Ta (endpoints of P1 ) : a ≤ Jj } where Jj = λ j +
j
(11λi−1 + 2νi +
i=2
2 1 1 τ +1 1 ( ) )ni ≤ ( )4(τ +1)µj−1 , κ δi−1 Bρj−1
1 1 (log((νi − 8λi−1 )(αi−1 − αi )) − log log( )), 2 log 2 εi−1 where B = B(κ, τ, s, dim L, β1 , γ1 , µ1 ). This implies that any interval of length κ ≤ τ Jj ni =
intersects at most p = #P1 many pieces of the partition Pj (λj ), i.e.  Pj (λj )  ≥
2 κ ≥ (κBρj−1 )4(τ +1)µj−1 . τ Jj
The third condition on ρj in (7.14) therefore becomes ρj ≤ (κBρj )
3 4τ (τ +1)µ2 j 4sµj+1 (µ1 ...µj+1 )6
which is fulﬁlled if for example µ21 ≥ 16sτ (τ + 1). We can now derive the following theorem.
40
L. H. Eliasson
Theorem 12. Let D ∈ N F(α, . . . , ρ; Ω, P)&T (σ, s) be covariant with respect to the quasiperiodic Laction (7.12), and assume that D is truncated at distance ν from the diagonal. Let F be a covariant matrix, smooth on P and  Fab C k ≤ εe−αb−a γ k ∀k ≥ 0. Then there exists a constant C – C depends only on dim L, κ, τ, s, α, β, γ, λ, µ, ν, ρ, σ, #P – such if ε ≤ C then there exists a matrix U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),
∀x ∈ X,
and D∞ (x) is a norm limit of normal form matrices Dj (x). Moreover D∞ (x) is pure point for a.e. x. The limit limj→∞ Ej (x) = E∞ (x) is uniform and satisﬁes, for all y ∈ / 2πZ, Lebesgue{x : E∞ (x + y) − E∞ (x) = 0} = 0. Moreover, if the Ej ’s and E∞ are real, then for all subsets Y −1 (Y ) = 0 Lebesgue(E∞
if
Lebesgue(Y ) = 0.
If D and F are Hermitian, then U is unitary and D∞ is Hermitian. Proof. Deﬁne the sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj by the formulas (7.8) and (7.1213), starting with α, β, γ, λ, µ, ν, ρ, σ, ε. Then they verify the assumptions (6.79), so we can apply Proposition 8, and (7.11), so we can apply Lemma 911. Use now Proposition 8(III). By Lemma 9–11 we get C(λj , µj , νj )clusteering for all j ≥ 1, hence statement (S ∞ ) holds. The ﬁrst part of the theorem now follows from Proposition 8(III) – it also gives the Hermitian case. In order to prove that D∞ (x) is pure point for a.e. x we must estimate the number ηj in Proposition 8(IV). This is the only place where we use condition (6.4). For each a in  a ≤ νj+1 + 2λj we consider uj (x, x + aω) = Res((Dj )Ωj (x+y) (x + aω), (Dj )Ωj (x) (x)) and the set  uj (x, x + aω)  ≤
ρj µ2
.
βj j
Using Lemma B1 we get that this this is a union of intervals of length less than 2 (νj+1 + 2λj )τ sµ12j ) . (2ρj Lj = γj σj κ The number of intervals does not exceed 2
Mj = 2sµj [
2 γj (νj+1 + 2λj )τ 8(βj µ2j )µj (sµ2j + 1)2 + #P)]. κσj
Perturbations of linear quasiperiodic system
41
This gives an upper bound for ηj , ηj ≤ const(νj+1 + 2λj )dim L Lj Mj and the veriﬁcation that
ηj < ∞
is straight forward (with the choice we have made of νj+1 ). The limit is uniform since  Ej+i (x) − Ej (x)  ≤  Dj+i (x) − Dj (x)  . Clearly, {x : E∞ (x+y)−E∞ (x) = 0} ⊂ {x : Ej (x+y)−Ej (x) ≤ D∞ (x)−Dj (x) } and this set is easy to estimate using the transversality on blocks of the matrix Dj . Suppose now that the Ej ’s and E∞ are real and let Y be a set of measure 0. We can use a covering ∪i Ii ⊃ Y with the following property: each Ii is contained in an interval I˜i , and each I˜i is contained in a piece of the partition Pji (λji ), ji ≥ J, in such a way that its distance to the boundary of I˜i is at least  D∞ (x) − Dji (x)  . Then Eji is smooth on I˜i and i
−1 Lebesgue(E∞ (Ii )) ≤
Lebesgue(Ej−1 (I˜i )). i
i
This sum is easy to estimate – it is essentially equal to its ﬁrst terms because of the rapid convergence – using the estimate of the resultants. Since we can −1 (Y ) is 0 if the measure do this for any J, this shows that the measure of E∞ of Y is 0. In the multilevel case the proofs goes through in the same way and all the results remain the same with obvious modiﬁcations. We can now complete the diagram of this perturbative process in Fig. 3.
42
L. H. Eliasson
ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 ) ⇓ ⇓
⇐⇐⇐
T (σ1 , s)
& ⇓
⇓
C(λ2 , µ2 , ν2 ) − clustering
⇓
⇓
⇓
ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 )
⇒⇒⇒ &
⇓ ⇓
⇐⇐⇐
T (σ2 , s) ⇓
⇓
C(λ3 , µ3 , ν3 ) − clustering
⇓
⇓
⇓
ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 )
⇒⇒⇒ &
⇓ ⇓
⇐⇐⇐
T (σ3 , s) ⇓
⇓
C(λ4 , µ4 , ν4 ) − clustering
⇓
⇓
⇓ Figure 3
8
Applications
8.1
Discrete Schr¨ odinger Equation
We consider the equation in strong coupling −ε(un+1 + un−1 ) + V (θ + nω)un = Eun ,
n ∈ Z,
(8.1)
where V is a real valued function on the onedimensional torus T = R/(2πZ) and ω is a real number. We assume that V is piecewise Gevrey smooth on a partition with p many pieces and  V C k ≤ βγ k
∀k ≥ 0.
(8.2)
The functions V (θ + y) − V (θ)) satisﬁes for all θ, y the two transversality conditions
max0≤k≤s  ∂yk (V (θ + y) − V (θ))  ≥ σ (8.3) max0≤k≤s  ∂θk (V (θ + y) − V (θ))  ≥ σ y . We assume also that ω is Diophantine, i.e. for some τ > 1, κ > 0 nω ≥
κ  n τ
∀n ∈ Z \ 0.
(8.4)
Theorem 13. Under the assumptions (8.24) there exist a small constant ε0 (p, β, γ, s, σ, κ, τ ) and, for all  ε ≤ ε0 , a function E∞ (θ) such that for almost all θ ∈ T and for all k ∈ Z, equation (8.1) has a solution uk ∈ l2 (Z) with E = E∞ (θ + kω). The set {uk : k ∈ Z} is an orthogonal basis for l2 (Z).
Perturbations of linear quasiperiodic system
43
If we identify the right hand side of (8.1) as an operator Hθ , then the statement says that Hθ is pure point for almost all θ. The spectrum of Hθ is the closure of Range(E∞ ) and it is easy to verify that the Lebesgue measure of [inf V, sup V ] \ Range(E∞ ) goes to 0 when ε goes to 0. Proof. We identify (8.1) with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = 1, β, γ, λ = 1, µ = 1, ν = 1, ρ = 1). Since max
0≤k≤sµ2

1 σ ∂yk (V (θ + y) − V (θ))  ≥ = σ ˜, 2 k (k!) γ (s!)2 γ s
and similar for the θderivative, we have also D ∈ T (˜ σ , s). The perturbation εF satisﬁes ∀k ≥ 0.  (εF )ba C k ≤ (εe)e−αa−b The theorem is now an immediate consequence of Theorem 12.
In weak coupling we consider the equation −(un+1 + un−1 ) + εV (θ + nω)un = Eun ,
n ∈ Z,
(8.5)
where V is a real valued function on the torus Td and ω ∈ Rd . We assume that V is analytic, satisfying sup  V (θ)  ≤ 1,
(8.6)
 θ d, κ > 0 < k, ω > ≥
κ  k τ
∀k ∈ Zd \ 0.
(8.7)
Theorem 14. Under the assumptions (8.67) there exist a constant ε0 (r, κ, τ ) and for all  ε ≤ ε0 a function E∞ (ξ) such that for almost all ξ ∈ T and for all k ∈ Zd , equation (8.5) has a solution ukn = ein(ξ+) U k (θ + nω) in l∞ (Z) with E = E∞ (ξ+ < k, ω >). The set {U k : k ∈ Zd } is an orthogonal basis for L2 (Td ). Proof. The equation for U ∈ L2 (Td ) is identiﬁed, in Fourier coeﬃcients, with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = r, β = 2, γ =
1 , λ = 1, µ = 1, ν = 1, ρ = 1). r
44
L. H. Eliasson
The eigenvalue is E(ξ) = cos(ξ) and the functions E(ξ + y) − E(ξ) satisﬁes the transversality conditions (8.3) for all ξ, y with some σ and s = 2. (Any nonconstant analytic function with primitive period 2π satisﬁes condition (8.3) for some σ, s.) Hence D ∈ T (˜ σ , 2). The Fourier coeﬃcients of V decays exponentially with the factor α = r which shows that εF satisﬁes the required smallness condition. The theorem is now an immediate consequence of Theorem 12. These two theorems claims nothing about exponential decay of the solutions uk or analyticity of the U k , i.e. exponential decay of its Fourier coeﬃcients. The uniform exponential decay of all the uk ’s is measured in the iteration process by the sequence αj which goes to 0, and must do so. If we instead let α vary over the partition so that we had one αvalue for each block, it follows from the proof that αj (x) stops decaying towards 0 as soon as the blocks Ωj (x) stops increasing. This will give exponential decay. (Exponential decay of eigenvectors have been proven in [3,4,6].) 8.2
Discrete Linear SkewProducts
We consider ﬁrst the weakly perturbed skewproduct of the form Xn+1 − (A + εB(θ + nω))Xn = EXn ,
n ∈ Z,
(8.8)
where A ∈ Gl(N, R), B : Td → gl(N, R). We assume that B is analytic, satisfying sup  B(θ)  ≤ 1
(8.9)
 θ). The set of functions {Y k,j : k ∈ in l∞ (Z) ⊗ CN with E = E∞ d Z , j = 1, . . . , N } is an orthogonal family in L2 (Td ) × CN .
Proof. Assume ﬁrst that A is semisimple. Since we construct complexvalued solutions we can without restriction assume that A and B are complex valued and that A is diagonal with diagonal elements a1 , . . . , aN . The equation for Y ∈ L2 (Td ) ⊗ CN is identiﬁed, in Fourier coeﬃcients, with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = r, β, γ =
1 , λ = 1, µ = 1, ν = 1, ρ = 1), r
Perturbations of linear quasiperiodic system
45
with β = A . It is now a multilevel matrix with eigenvalues E j (ξ) = eiξ − aj ,
j = 1, . . . , N
and the blocks Ωj = {0} × {j}, j = 1, . . . , N . The functions E i (ξ + y) − E j (ξ) satisfy the transversality conditions (8.3) for all ξ, y with some σ and s = 1. Hence D ∈ T (˜ σ , 1). The Fourier coeﬃcients of B decays exponentially with the factor α = r which shows that εF satisﬁes the required smallness condition. The theorem is now an immediate consequence of Theorem 12 – multilevel version. The case when A is not semisimple is almost the same. The matrix D is still on normal form, D ∈ N F(α = r, β, γ =
1 , λ = N, µ = N, ν = 1, ρ = 1), r
with β = A . It is still a multilevel matrix with eigenvalues E j (ξ) = eiξ − aj ,
j = 1, . . . , N
but with the blocks Ωj = {0} × {1, . . . , N }, j = 1, . . . , N , reﬂecting that it is block diagonal instead of diagonal. The transversality of the functions E i (ξ +y)−E j (ξ) implies D ∈ T (˜ σ, N 2 ) through Lemma B2. Now we can apply theorem 12. The right conjecture about equation (8.8), under conditions (8.7,9), is the following: for E = 0 (or for any ﬁxed E), if ε is small enough then (8.8) has a Floquet representation for almost all matrices A. This means that the fundamental solutions can be written as en∆ Z(θ + nω),
n ∈ Z,
where Z : Td → Gl(N, R) and ∆ ∈ gl(N, R). Theorem 15 and its proof is probably not so far from proving this conjecture. An even stronger result may be true: instead of “almost all matrices A” we may put “almost all matrices in any “generic” oneparameter family At ”. Such a result has been proven for certain oneparameter families in Sl(2, R) [13] and in compact groups [17,18]. The application to the strongly perturbed skewproduct (1.4) is straight forward, but the continuous Schr¨ odinger in one dimension in strong coupling (1.5) is more delicate. Theorem 12 will not apply to the discrete equation (1.6), derived from (1.5) in the introduction, since W will not satisfy the full transversality condition. It seems however that W is transversal in a neighborhood of W −1 (0) when E is close to inf V , if V is transversal near inf V . We therefore believe that when ε small enough and when E is suﬃciently close to inf V , then the spectrum of the operator (1.6) is pure point near 0 – this has been proven in some cases [4].
46
L. H. Eliasson
A
Appendix A
Gevrey classes Let uj , j = 1, 2 be smooth functions in the Gevrey class G2 , deﬁned in a convex set V in Rd , and consider the norm  uj C =:
sup sup  0≤l≤k x∈V
1 l ∂ uj (x)  (l!)2
– here k and l are multiindices in Nd . Lemma A.1 Assume that  uj C k ≤ βj γ k , ∀k ≥ 0. Then for all k ≥ 0 (i)  u1 u2 C k ≤ 4d (β1 β2 )γ k ; (ii)  eu1 C k ≤ e4 (iii) 
d
β1 k
γ ;
√ 1 + u1 C k ≤ (1 + β1 )γ k
if
4d β1 < 1;
(iv) 
1 4d β1 k 1 γ) C k ≤ ( u1 δ δ
if
 u1 > δ.
Proof. k ∂ k−l u1 ∂ l u2  l 0≤l≤k k (l!)2 ((k − l)!)2 ≤ (β1 β2 )γ k (k!)2 , l (k!)2
 ∂ k (u1 u2 )  = 
0≤l≤k
and the result follows since d ki −1 k −1 = [ ] ≤ 4d l l i i=1
0≤l≤k
0≤li ≤ki
for all k – just notice that klii ≥ 2li if li ≤ k2i . √ We use the power series expansions of eu and 1 + u which converges absolutely in  u < ∞ and in  u < 1 respectively, to prove the second and third estimate.
Perturbations of linear quasiperiodic system
47
Notice that  u1 > δ implies that β1 > δ. By a scaling we can suppose that δ = 1 and β1 ≥ 1. The fourth statement is obvious for k = 0 so we proceed by induction on  k . Hence k k−l 1 l 1 k 1 ∂ ( )∂ u1  ∂ ( ) =− l u1 u1 u1 0≤l≤k, l=0 k (l!)2 ((k − l)!)2 d(k−1) 2 k ≤ 4 (k!) (β1 γ) l (k!)2 0≤l≤k
≤ (k!) (4 β1 γ) . 2
d
k
Remark. The preceding lemma is valid also for matrices if we understand by u1 the inverse of u, and if we use the operator norm satisfying  uv ≤ u  v  . Estimates of eigenvalues Let D(x) be a m × mmatrix, smooth in a convex set V in Rd containing 0, and let {E j (x)}m 1 be a continuous choice of its eigenvalues. Lemma A.2 (i)  E j (x)  ≤  D(x)  . (ii) 1
1
 E j (x) − E j (x )  ≤ 4  D C 0 m  ∂D Cm0  x − x  m . ¯ then (iii) If D is Hermitian, i.e. Dt = D, 1−
1
 E j (x) − E j (x )  ≤  ∂D(x) C 0  x − x  . Proof. (i) is obvious. To see (ii), let I be the segment joining x and x and let δ = E 1 (x) − E 1 (x ) . Then there is an x ∈ I such that, δ  P (λ, x)  ≥ ( )m , λ = E 1 (x ), 4 m j where P (λ, x) = i=1 (λ − E (x)). (This is a wellknown inequality of Chebyshev – see e.g. [28].) Hence δ ( )m ≤  P (λ, x) − P (λ, x )  ≤  ∂x P (λ, ) C 0  x − x  . 4 Since  ∂x P  ≤
m j=1
 det(D1 . . . ∂x Dj . . . Dm )  ≤
m i,j=1
 ∂Dij  D m−1
48
L. H. Eliasson
and  ∂Dij ≤ m  ∂D  the result follows. To see (iii) let q j (x) be the eigenvector corresponding to E j (x). Then (D(x) − E j (x)I)q j (x) = 0, and if we diﬀerentiate the relation and take the scalar product with q j (x) and use that the eigenvectors are orthogonal, then we get an estimate of ∂E j . When Ej and q j are not diﬀerentiable one uses the same argument with a diﬀerence operator. Better estimates requires separation of eigenvalues. Assume that  D C k ≤ βγ k
∀k ≥ 0.
Lemma A.3 Assume that the eigenvalues belong two groups
E 1 (x), . . . , E n (x) E n+1 (x), . . . , E m (x) such that any eigenvalue of one group is separated from any eigenvalue of the other group by at least r for all x ∈ V . Then the polynomial n
(λ − Ej (x)) =
j=1
n
en−j (x)λj
j=0
is smooth in V and satisﬁes n j βm m k β ((const  ej C k ≤ ) γ) j r
∀k ≥ 0.
If D is Hermitian, then we also have n j β jγ.  ∂ej C 0 ≤ j The constants are independent of m. Proof. Notice that r ≥ 2β, and that the supestimate of the coeﬃcients and, in the Hermitian case, of the ﬁrst derivative of the coeﬃcients follows from Lemma A.2. By scaling we can assume that β = 1 if we replace r by βr . The polynomial P (λ, x) = det(λI − D(x)) satisﬁes  P (λ, ) C k ≤ ((const)m γ)k if just  λ < 2.
∀k ≥ 0,
Perturbations of linear quasiperiodic system
49
Choose a curve ∆(x) in  λ ≤ 1 + 12 – piecewise constant in x – keeping a distance ≥ 2r to all the roots E 1 (x), . . . , E m (x) of P (λ, x) and surrounding the ﬁrst n of these roots. ∆(x) may consist of several components so we can choose it to be of length at most nπr. Then we have for all λ ∈ ∆(x)
 P (λ, x)  ≥ ( 2r )m = ρ (λ,x)  ∂λPP(λ,x)  ≤ 2m r . Using this we verify, as in Lemma A.1, that 
2m 1 ∂λ P (λ, x) C k ≤ ((const)m γ)k P (λ, x) r ρ
∀k ≥ 0
– where we have used the Cauchy formula to estimate  ∂λ P (λ, ) C k on  λ ≤ 1 + 12 . Consider now the power symmetric functions in the ﬁrst n roots: pj (x) = E 1 (x)j + · · · + E n (x)j . We have the integral representation 1 ∂λ P (λ, x) dλ, λj pj (x) = 2π ∆(x) P (λ, x) from which we get 1  pj C k ≤ mn2j ((const)m γ)k ρ
∀k ≥ 0.
The ej ’s are the elementary symmetric functions in we have the relation p1 1 0 0 p2 p1 2 0 (−1)j p p1 3 ej = det 3 p2 . j! .. .. .. . . . . .
the ﬁrst n roots and ... . . . . . . , .. .
pj pj−1 pj−2 pj−3 . . . for j ≥ 1 [29, page 20] – of course e0 = 1. So j!ej is a sum of at most 2j many products pι1 · · · · · pιl , ι1 + · · · + ιl = j ≤ m, each with a coeﬃcient that is at most (j − 1) · (j − 2) · · · · · l. From Lemma A.1 we now get the estimate we want. Orthogonalization Let v 1 , . . . , v m be vectors in Cn which are smoothly deﬁned in a convex neighborhood V of 0 in Rd . Assume that v 1 (0), . . . , v m (0) is ON and that  v j C k ≤ βγ k
∀k ≥ 0,
for j = 1, . . . , m. (This implies in particular that β ≥ 1.)
50
L. H. Eliasson
Lemma A.4 Then there is a constant, independent of m, and an upper triangular m × nmatrix R, R(0) = I, smooth in W = {x : x 
C k ≤ const i = j m const i i < v , v > −1 C k ≤ m in  x < const mβγ . In particular they remain linearly independent and of norm ≤ 2 if the constant is small enough. Let now m−1 vˆm = v m − ai v i i=1
and determine the ai ’s so that < vˆm , v >= 0 for all i = 1, . . . , m − 1. This gives a system of linear equation in the ai ’s which we can solve by Gauss elimination if the constant is small enough (independent of m). The solution satisﬁes
k  ai C k ≤ const i = 1, . . . , m − 1 m γ m k  vˆ C k ≤ constβγ i
for all k ≥ 0. If the constant is small enough we get  vˆm  ≥ and we can estimate wm = tion.
v ˆm ˆ vm 
1 2
by Lemma A.1. Then we proceed by induc
Subspaces and Angles Assume now that Λ1 and Λ2 are two subspaces in Cm with Λ1 ∩ Λ2 = {0}, and recall that they come equipped with ONframes
Perturbations of linear quasiperiodic system
51
Sublemma. The spectrum of the matrices Λ∗1 Λ2 Λ∗2 Λ1 and Λ∗2 Λ1 Λ∗1 Λ2 only diﬀer by an eigenvalue 0 of multiplicity =  dim Λ1 − dim Λ2 . The spectrum of I − Λ∗1 Λ2 Λ∗2 Λ1 is contained in [0, 1]. Proof. Notice that if λ = 0 is an eigenvalue of AB, then it is also an eigenvalue of BA, with the same multiplicity. Hence, the spectrum of AA∗ and A∗ A is the same except possibly for an eigenvalue 0. Now the kernel of A∗ A is precisely the kernel of A, so if A is an k × lmatrix of rank r then the dimension of the kernel is l − r. Since the rank of A and A∗ is the same, we get that 0 is an ∗ eigenvalue of AA and A∗ A of multiplicity k − r and l − r respectively. I A and Λ2 = , then If now Λ1 = 0 B Λ∗1 Λ2 Λ∗2 Λ1 = AA∗ ,
Λ∗2 Λ1 Λ∗1 Λ2 = A∗ A,
so the ﬁrst ﬁrst statement is proved. Since Λ∗2 Λ2 = A∗ A+B ∗ B, we know that I −A∗ A = B ∗ B. Now any matrix of the form B ∗ B is positive deﬁnite which implies that spectrum of I − A∗ A belongs to [0, ∞[ and to ] − ∞, 1]. This proves the second statement. We can now deﬁne the angle between Λ1 and Λ2 as the unique ϕ ∈ [0, π[ such that dist(σ(Λ∗1 Λ2 Λ∗2 Λ1 ), 1) = sin(ϕ). If these spaces are onedimensional, this notion coincides with the “usual” angle between vectors. Consider now an m × mmatrix D,  D ≤ β, whose eigenvalues belongs to two groups that are separated by a distance at least r. Let Λ1 and Λ2 be the invariant subspaces corresponding to the two groups. Lemma A.5 The angle between the spaces Λ1 and Λ2 ≥ (const
1 m+1 r (m+1) m 2 , ) 2 ( ) m β
if r < constβ. Proof. We can assume that
D=
S A 0 T
,
where S and T are upper triangular with the diagonal elements of one separated by r from those of the other. We now look for an m1 × m2 matrix R, m = m1 + m2 , m2 ≤ m1 , such that −1 I R I R S 0 D = . 0 I 0 I 0 T
52
L. H. Eliasson
I
Then the vectors
0
R
and
I
will span Λ1 and Λ2 respectively. For a matrix M denote by Mi the matrix whose (k, l)entry is Mkl if l−k = i and 0 otherwise. Then S = S0 + · · · + Sm1 ,
T = T0 + · · · + Tm2 ,
A = A−m1 + · · · + Am2 .
The equation for R, SR − RT = −A, can now be written S0 Rj − Rj T0 = −Aj −
Sk Rj−k +
k≥1
Rj−l Tl
l≥1
for −m1 ≤ j ≤ m2 . From this we get that β  R  ≤ constm( )m+1 . r It follows from this by GramSchmidt that we get an ONbasis for Λ2 A B where B is a triangular m2 × m2 matrix whose diagonal entries have absolute value 1 r ≥ const ( )m+1 , m β and  B  satisﬁes the same estimate as  R . If now v ∈ Cm2 with  v = 1, then Av 2 1 ≥  =  Av 2 +  Bv 2 , Bv which implies that  A 2 ≤ 1 − [(const Since
1 r m+1 m2 2 ( ) ) ] . m β
 A∗ A  = A , it follows that  A∗ A  ≤ 1 − [(const
1 r m+1 m2 2 ( ) ) ] , m β
which bounds the spectrum of A∗ A. Since sine of the angle equals the square root of dist(σ(A∗ A), 1), the result follows.
Perturbations of linear quasiperiodic system
53
Block splitting Let D be an m × mmatrix, smooth in a convex neighborhood V of 0 in Rd , and assume that  D C k ≤ βγ k ∀k ≥ 0. Lemma A.6 For any r > 0 there exists a D(x)invariant decomposition Cm =
k
Λj (x),
i=1
smooth on W = { x ≤ (const
r 2m 1 ) }∩V βm γ
with the following properties: (i) all eigenvalues of DΛj are rclose, i.e. for any given x E and E are eigenvalues of the same block D(x)Λj (x) ⇒
 E − E ≤ r;
(ii) βm 2m k ) γ) r (iii) the angles between the diﬀerent Λj ’s are  Λj C k ≤ ((const
≥ (const
∀k ≥ 0;
r m2 ) ; βm
(iv) the matrix Q = (Λ1 . . . Λk ) satisﬁes  Q−1 C k ≤ (const
βm 2m2 βm 4m2 k ) ) ((const γ) r r
∀k ≥ 0.
All the constants are independent of m. Proof. We can take β = 1 if we replace r by βr . Make now a decomposition r separated groups as possible, and of the eigenvalues of D(0) into as many m consider the corresponding decomposition of Cm into invariant subspaces 1 m Λj (0). Choose a basis v (0), . . . , v (0) which is ON in each Λj (0). r By Lemma A.2 the groups of eigenvalues of D(x) remain 2m separated for x in r m1 ) } ∩ V. U = { x  ≤ (const βm γ ˇ (λ−E i (x)), Let {E j (x)}m j=1 be the eigenvalues of D(x) and let Pj (λ, x) = where the product is over all eigenvalues except those associated to Λj (0) – by Lemma A.3 we have estimates of the coeﬃcients of Pˇj (λ, x).
54
L. H. Eliasson
For v i (0) in Λj (0), let 1 Pˇj (D(x), x)v i (0), i ˇ Pj (E (0), 0)
v i (x) =
x ∈ U.
This gives a basis for Λj (x) for x near 0 – how near? We have m m  v i C k ≤ ( )m−1 (const)m ((const )m γ)k ∀k ≥ 0. r r Hence, for x ∈ W , v i (x) will be close to v i (0) and we get a basis deﬁned in W . Lemma A.4 gives an ONbasis for each Λj (x) on W which fulﬁlls the estimate. The estimate of the angle follows from Lemma A.5, using that m2 ≥ (m + 1) m 2. In order to estimate the inverse of Q, consider the D∗ invariant decomposition k ˜ j (x), Cm = Λ i=1
such that the eigenvalues of D∗ (x)Λ˜ j (x) are complex conjugates of the eigen˜ j satisﬁes the same estimate as Λj and Λ ˜j = values of D(x)Λj (x) . Then Λ ⊥ ( i=j Λi ) . If we let ˜ = (. . . Λ ˜ iΛ ˜j . . . ) Q then
B
−1
∗ ˜ =: Q Q =
..
.
˜ i )∗ Λi (Λ ˜ j )∗ Λj (Λ ..
.
˜∗. so that Q−1 = B Q ˜ j )⊥ is at least The angle between Λj and ( i=j Λi ) = (Λ (const
r m2 ) . m
Therefore ˜ j )∗ Λj = I − (Λj )∗ (Λ ˜ j )⊥ ((Λ ˜ j )⊥ )∗ Λj ˜ j (Λ Sj =: (Λj )∗ Λ is a symmetric matrix with spectrum bounded away from 0 by at least the square of that angle. Hence, by Lemma A.1, 2 2 m m  Sj−1 C k ≤ (const )2m ((const )4m γ)k ∀k ≥ 0. r r −1 ∗ −1 ∗˜ ˜ Since ((Λj ) Λj ) = S (Λj ) Λj we get the same estimate for B as for S −1 j
and, hence, the same estimate for Q−1 as for Sj−1 .
j
Perturbations of linear quasiperiodic system
55
Remark. When D is Hermitian in Lemma A.6 the angle between the invariant spaces is of course π2 and Q is unitary so  Q−1 C k ≤ ((const
βm 2m k ) γ) r
∀k ≥ 0.
Truncation Let D : l2 (L) → l2 (L) be an inﬁnitedimensional matrix with a basis of ﬁnitedimensional (generalized) eigenvectors {q b } and blocks Ωb . Suppose that D∗ also has a basis of (generalized) eigenvectors q˜b with the same blocks Ωb . We enumerate the eigenvectors so that q b and q˜b correspond to complex conjugate eigenvalues with the same multiplicity. This implies that q˜b ⊥ q a , for all a = b. Lemma A.7 Let Ω = (Ωb ) = ∪Ωa ∩Ωb Ωa ,
b ∈ L.
If v is an (generalized) eigenvector of DΩ then either v is an (generalized) b eigenvector of D or v ⊥ CΩ . ∗ Proof. Notice that q a and q˜a are (generalized) eigenvectors of DΩ and DΩ a b respectively, if Ω ∩ Ω = ∅. Let now v be an (generalized) eigenvector of DΩ . Then either v = q a for a some a such that Ωa ∩ Ωb = ∅ – and wec are done – or v ⊥ q˜ for all a such a b that Ω ∩ Ω = ∅. Since now v = Ac q we get
0 = < q˜a , v > = Aa < q˜a , q a >, which implies that Aa = 0. This means that v = Ac q c Ωc ∩Ωb =∅ b
so v is perpendicular to CΩ .
A numerical lemma Lemma A.8 a0 =a,a1 ,...,aj =b
√ 6 e−(a−a1 +···+aj−1 −b)s ≤ ( dim L )(j−1) dim L s
∀a, b ∈ L.
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L. H. Eliasson
Proof. If we take  ·  as the l1 norm we get e−(a−a1 +···+aj−1 −b)s = a0 =a,a1 ,...,aj =b
e−((b−a)−c1 −···−cj−1 +c1 +···+cj−1 )s ≤
c1 ,...,cj−1
(
e−( j−1 −c+c) 3 )j−1 ≤ ( b−a
s
c
s 6 b−a 6  + )(j−1)d e−b−a 3 ≤ ( )(j−1)d j−1 s s
(d = dim L), where we used the inequality d−
j−1
ci  +
j−1
1
1
1 d − ci  +  ci ). ( 3 1 j−1 j−1
 ci  ≥
This inequality is easy to verify because the 3×RHS is less then d  +2  ci ) = d  +2  ci ≤ j−1 d− ci  +  ci  +2  ci ≤ d − ci  +3  ci ,
(
where summation √ goes from 1 to j − 1. The factor dim L comes in because we use th l2 norm.
B
Appendix B
Estimates of preimages Let u be a real or complex valued smooth function deﬁned on an open interval ∆ in R and satisfying
 u C k ≤ βγ k ∀ 0≤k ≤m+1 1 k max  (k!)2 γ k ∂ u(x)  ≥ σ ∀x ∈ ∆. 0≤k≤m
Lemma B.1 There exists a ﬁnite union of open intervals ∪j∈J ∆j such that 2 ≤ 2m [ 8βγ(m+1)  ∆  +1] #J σ 1 2 2ρ m maxj∈J  ∆j  ≤ γ ( σ )  u(x)  ≥ρ ∀x ∈ ∆ \ ∪∆j . Proof. Assume for simplicity that 0 ∈ ∆ and that  ∂ m u(0) ≥ σ(m!)2 γ m . ˜ of length Then there is an interval ∆ σ 8βγ(m + 1)2
Perturbations of linear quasiperiodic system
57
˜ if u is real – if u is complex valued such that  ∂ m u(x) ≥ σ2 (m!)2 γ m on ∆, then this holds for u or u. ˜ There exists an interval ∆1 of length Consider now ∂ m−1 u on ∆. ≥
2 2ρ 1 ( )m γ σ
such that σ 2ρ 1 ( ) m ((m − 1)!)2 γ m−1 2 σ
 ∂ m−1 u(x) ≥
˜ \ ∆1 . ∀x ∈ ∆
˜ \ ∆1 . There exist two intervals ∆2 , ∆3 , each of Consider now ∂ m−2 u on ∆ length 2 2ρ 1 ( )m , γ σ such that  ∂ m−2 u(x) ≥
σ 2ρ 2 ( ) m ((m − 2)!)2 γ m−2 2 σ
˜ \ ∆ 1 ∪ ∆2 ∪ ∆3 , ∀x ∈ ∆
etc. ˜ 2m − 1 (possibly void) intervals ∆i such that Hence we obtain in ∆,  u(x)  ≥ ρ  ∆i  ≤
˜ \ ∪∆i , ∀x ∈ ∆ 2 2ρ 1 ( )m . γ σ
On the whole interval ∆ we get at most 2m × [ many such intervals.
8βγ(m + 1)2  ∆  +1] σ
Transversality of products of functions Let now uj be a sequence of real or complex valued smooth functions deﬁned on an open interval ∆ in R and satisfying
∀ 0 ≤ k ≤ m + 1 = m1 + · · · + mj + 1  uj C k ≤ βγ k max  (k!)12 γ k ∂ k uj (x)  ≥ σ ∀x ∈ ∆. 0≤k≤mi
Lemma B.2 If u = u1 · · · · · uj , then
 u C k ≤ 4d(j−1) β j γ k 2 max  (k!)12 γ k ∂ k u(x)  ≥ ( 12 )8m σ j(m+1)
0≤k≤m
∀ 0 ≤ k ≤ m + 1, ∀x ∈ ∆.
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Proof. The ﬁrst part follows from Lemma A.1 so we concentrate on the second estimate. We can assume that β = 1 and γ = 1 if we replace the ui ’s by uβi and rescale x. Fix an x – x = 0 say. We can assume without restriction that 1  ∂ mi ui (0)  ≥ σ (mi !)2
∀i.
Order the mi ’s in decreasing order m1 ≥ m2 ≥ . . . . Fix a symmetric interval I around 0 of length 2δ ≤
σ . 8(m + 1)2
By Lemma B1 we get that  u1 ≥ ρ outside a subset of I of measure less than 2ρ 1 N = 2m1 +2 , N ( ) m1 , σ which is less than δj if we chose ρ =
σ δ m1 ( ) . 2 jN
Applying this to all the ui ’s we get that σ δ m )  u(x)  ≥ ρj = ( )j ( 2 jN outside a set of measure less than δ. Consider now the Taylor expansion u(x) = a0 + a1 x + · · · + am xm + u ˆ(x). Then, for  x ≤ δ, u ˆ(x)  ≤ (m + 1)!  x m+1 ≤ (m + 1)!δ m+1 . From this we get an
δ 2
≤ x ≤ δ such that
 a0 + a1 x + · · · + am xm  ≥ if we choose δ =
1 σ j δ m ( ) ( ) , 2 2 jN
σ 1 1 m ( )j ( ) . 2(m + 1)! 2 jN
Hence for δ so small and some  ak xk  ≥
δ 2
≤ x ≤ δ and some 0 ≤ k ≤ m, σ 1 δ m ( )j ( ) . 2(m + 1) 2 jN
Since k!ak = ∂ k u(0)the result follows.
Perturbations of linear quasiperiodic system
59
Remark. In [5] there was given a short proof of this lemma with a worse estimate, which however is good enough for the applications we have in mind. The above proof, which is just as short but gives a considerable better estimate, is due to M. Goldstein [6]. Acknowledgement. This paper was partially written during a visit to IMPA, Rio de Janeiro, ﬁnanced by the STINT foundation.
References 1. Craig, W.: Pure point spectrum for discrete almost periodic Schr¨ odinger operators. Commun. Math. Phys. 88, 113–131 (1983) 2. P¨ oschel, J: Examples of discrete Schr¨ odinger operators with pure point spectrum. Commun. Math. Phys. 88, 447–463 (1983) 3. Sinai, Ya.G.: Anderson localization for the onedimensional diﬀerence Schr¨ odinger operator with a quasiperiodic potential. J. Stat. Phys. 46 861–909 (1987) 4. Fr¨ ohlich, J., Spencer, T., Wittver, P.: Localization for a class of onedimensional quasiperiodic Schr¨ odinger operators. Commun. Math. Phys. 132, 5–25 (1990) 5. Eliasson, L.H.: Discrete onedimensional quasiperiodic Schr¨ odinger operators with pure point spectrum. Acta Math. 179, 153–196 (1997) 6. Goldstein, M.: Anderson localization for quasiperiodic Schr¨ odinger equation. Preprint, 1998. 7. Chulaevsky, V.A., Dinaburg, E.I.: Methods of KAMTheory for LongRange QuasiPeriodic Operators on Zν . Pure Point Spectrum. Comm. Math. Phys. 153, 559–577 (1993) 8. Avron,J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6, 81–85 (1982) 9. F. Delyon, D. Petritis: Absence of localization in a class of Schr¨ odinger operators. Commun. Math. Phys. 103, 441–443 (1986) 10. Dinaburg, E.I., Sinai, Ya.G.: The onedimensional Schr¨ odinger equation with quasiperiodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975) 11. R¨ ussmann, H.: On the onedimensional Schr¨ odinger equation with a quasiperiodic potential.In: Nonlinear Dynamics (Internat. Conf., New York, 1979), Helleman, R.H.G. (ed.) New York: New York Acad. Sci., 1980 12. Moser, J., P¨ oschel, J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helvetici 59, 39–85 (1984) 13. Eliasson, L.H.: Floquet solutions for the onedimensional quasiperiodic Schr¨ odinger equation. Commun. Math. Phys. 146, 447–482 (1992) 14. Bellissard, J., Lima, R., Testard, D.: A metal insulator transition for the almost Mathieu equation. Commun. Math. Phys. 88, 207–234 (1983) 15. Albanese, C.: KAM theory in momentum space and quasiperiodic Schr¨ odinger operators. Ann. Inst. H. Poincar´e, Anal. Non Lin´eaire 10, 1–97 (1993) 16. Goldstein, M.: Laplace transform methods in the perturbation theory of spectrum of Schr¨ odinger operators I and II. Preprints, 1991–92 17. Krikorian, R.: R´eductibilit´e presque partout des syst`emes quasi p´eriodiques analytiques dans le cas SO(3). C. R. Acad. Sci. Paris 321, S´erie I, 1039–1044 (1995)
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18. Krikorian, R.: R´eductibilit´e des syst`emes produits crois´es quasip´eriodiques ` a ´ valeurs dans des groupes compacts. Paris: Thesis Ecole Polytechnique, 1996 19. Eliasson, L.H.: Ergodic skewsystems on SO(3, R). Preprint ETHZ¨ urich, 1991 20. Eliasson, L.H.: Reducibility, ergodicity and point spectrum. Proceedings of the International Congress of Mathematicians. Berlin 1998, Vol. II, 779–787 (1998) 21. Young, L.S.: Lyapunov exponents for some quasiperiodic cocycles. Ergod. Th.& Dynam. Syst. 17, 483–504 (1997) 22. Chualevsky, V.A., Sinai, Ya.G.: Anderson localization and KAMtheory. In: P. Rabinowitz, E. Zehnder (eds.): Analysis etcetera. New York: Academic Press, 1989 23. Chualevsky, V.A., Sinai, Ya.G.: The exponential localization and structure of the spectrum of 1D quasiperiodic discrete Schr¨ odinger operators. Rev. Math. Phys. 3, 241–284 (1991) 24. Goldstein, M.: Quasiperiodic Schr¨ odinger equation and analogue of the Cartan’s estimate for real algebraic functions. Preprint, 1998 25. Eliasson, L.H.: Communication at the 4th Quadriennial International Conference on Dynamical Systems, IMPA, July 29–August 8, 1997 26. Bourgain, J.: Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schr¨ odinger equation. Ann. of Math. 148, 363–439 (1998) 27. Folland, G.D.: Introduction to partial diﬀerential equations. Princeton, N.J.: Princeton University Press, 1976 28. Shapiro, H.: Topics in approximation theory, LMN 187. Berlin: SpringerVerlag, 1971 29. MacDonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1979
KAMpersistence of ﬁnitegap solutions Sergei B. Kuksin
Introduction The paper is devoted to a proof of the following KAM theorem: most of spaceperiodic ﬁnitegap solutions of a Laxintegrable Hamiltonian PDE persist under small Hamiltonian perturbations as timequasiperiodic solutions of the perturbed equation. In order to prove it we obtain a number of results, important by itself: a normal form for a nonlinear Hamiltonian system in the vicinity of a family of lowerdimensional invariant tori, etc. The paper is an abridged version of the manuscript [KK], missing proofs can be found there. Still our presentation is rather complete modulo a KAMtheorem for perturbations of linear equations (for its proof see [K, K2, P]). Notations. Everywhere in the paper “linear map” means “bonded linear map”. For a linear operator J we denote by J¯ the operator −J −1 . For a complex Hilbert space, ·, · stands for a complexbilinear quadratic form such that u, u ¯ = u2 . We systematically use the following nonstandart notations: for a vector V = (V1 , . . . , Vn ) ∈ Nn we denote NV = {m ∈ N  m = Vj ∀ j} and ZV = {m ∈ Z  m = ±Vj ∀ j}; we abbreviate N(1,2,...,n) to Nn , etc. In particular, Z0 stands for the set of nonzero integers.
1 1.1
Some analysis in Hilbert spaces and scales Smooth and analytic maps
Below we shall work with diﬀerentiable maps between domains in Hilbert (more general, Banach) spaces. Since the category of C r smooth Fr´echet maps with r ≥ 2 is rather cumbersome and since only analytic object arise in our main theorems, we mostly restrict ourselves to the two extreme cases: with few exceptions the maps will be either C 1 smooth or analytic. Now we ﬁx corresponding notations and brieﬂy recall some properties of C 1 smooth and analytic maps. Let X, Y be Hilbert spaces and O be a domain in X. A continuous map f : O → Y is called continuously diﬀerentiable, or C 1 smooth (in the sense of Fr´echet) if there exists a bounded linear map f∗ (x) : X → Y which continuously depends on x ∈ O, such that f (x + x1 ) − f (x) = f∗ (x)x1 + o(x1 X ) provided that x, x + x1 ∈ O. We call f∗ (x) a derivative of f or its tangent map. By f ∗ (x) we denote the adjoint map f ∗ (x) = (f∗ (x))∗ : Y → X. For a real Hilbert space X we denote by X c its complexiﬁcation , X c = X ⊗R C .
L.H. Eliasson et al.: LNM 1784, S. Marmi and J.C. Yoccoz (Eds.), pp. 61–123, 2002. c SpringerVerlag Berlin Heidelberg 2002
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Examples. If X is an L2 space or a Sobolev space of realvalued functions, then X c is a corresponding space of complex functions. If X is an abstract separable Hilbert space and {φj } is its Hilbert basis, then X = { xj φj  xj ’s are real and xj 2 < ∞}, while X c = { zj φj  zj ’s are complex and . . . }. Let X c , Y c be complex Hilbert spaces and Oc be a domain in X c . A map f : Oc → Y c is called Fr´echetanalytic if it is C 1 smooth in the sense of real analysis (when we treat X c , Y c as real spaces) and the tangent maps f∗ (x) are complexlinear. Locally near any point in Oc such a map can be represented as a normally convergent series of homogeneous maps (see [PT]). For real Hilbert spaces X, Y and a domain O ⊂ X, a map F : O → Y is analytic if it can be extended to a complexanalytic map F : Oc → Y c , where Oc is a complex neighbourhood of O in X c . A map F : X ⊃ O → Y is called δanalytic (δ is a positive real number) if it extends to a bounded analytic map (O + δ) → Y c (O + δ is the δneighbourhood of O in X c ). We note that compositions of analytic maps are analytic, as well as their linear combinations. Besides, any analytic map is C k smooth for every k. There is an important criterion of analyticity: a map f : X c ⊃ Oc → Y c is analytic if and only if it is locally bounded 1 and weakly analytic, i.e., for any y ∈ Y c and any aﬃne complex plane Λ ⊂ X c the complex function Λ ∩ Oc → C, λ → F (λ), yY is analytic in the sense of one complex variable. Even more, it is suﬃcient to check analyticity of these functions for a countable system y = y1 , y2 , . . . of vectors in Y such that the linear envelope of this system is dense in Y (see [PT]). The Cauchy estimate states that if a map F : X c ⊃ Oc → Y c admits a bounded analytic extension to Oc + δ, then for any u ∈ Oc one has: F∗ (u)X,Y ≤ δ −1
sup
u ∈O c +δ
F (u )Y .
(The estimate readily follows from its onedimensional version applied to the holomorphic functions Oδ (C) λ → F (u + λx), yY , where xX = yY = 1). In particular, this estimate applies to δanalytic maps between subsets of real Hilbert spaces. If F : X c ⊃ Oc → Y c is an analytic map and for some point x ∈ Oc the tangent map F∗ (x) is an isomorphism, then by the inverse function theorem in a suﬃciently small neighbourhood of x the map F can be analytically inverted. The same is true for real analytic maps. See [PT]. For Banach spaces everything is much the same with one extra diﬃculty: there is no canonical way to deﬁne a complexiﬁcation X c of a real Banach space X . This diﬃculty should not worry us since all Banach spaces used 1
that is, any point x ∈ Oc has a neighbourhood, where f is uniformly bounded. In particular, any continuous map is locally bounded.
KAMpersistence of ﬁnitegap solutions
63
below are natural and one can immediately guess the right complexiﬁcation. For example, if X is the space of bounded linear operators Y1 → Y2 where Y1 , Y2 are Hilbert spaces, then X c is the complex space of linear over reals operators Y1 → Y2c , etc. 1.2
Scales of Hilbert spaces and interpolation
Below we shall study onedimensional in space partial diﬀerential equations in appropriate scales of Sobolev spaces {Xs }, where s ∈ Z or s ∈ R. That is, (i) each Xs is a Hilbert subspace of a Sobolev space of order s, formed by scalar or vectorfunctions on a segment [0, T ] (so X0 ⊂ L2 ); (ii) there exist: (1) a Hilbert basis {φk  k ∈ Z0 }, Z0 = Z \ {0}, of the space X0 , (2) an even positive sequence {ϑj  j ∈ Z0 } of linear growth, i.e. ϑj = ϑ−j and C −1 k ≤ ϑk ≤ Ck for all k, such that for any real s the set of vectors {φk ϑ−s k  k ∈ Z0 } form a Hilbert basis of the space Xs . The second assumption implies that for any −∞ < a < b < ∞ the space Xc , c = (1 − θ)a + θb, interpolates the spaces Xa and Xb : in notations of [LM], Xc = [Xa , Xb ]θ . In particular, for any u ∈ Xb holds the interpolation θ inequality: uc ≤ u1−θ a ub . The basis {φk } is called a basis of the scale. The norm and the scalar product in Xs will be denoted · s and ·, ·s = ·, ·Xs ; we abbreviate ·, ·0 = ·, ·. Due to ii), u, u2s = u2s = uk 2 ϑ2s if u = uk φk . k The scalar product ·, · extends to a bilinear pairing Xs × X−s → R. For any space Xs (real or complex) we identify its adjoint (Xs )∗ with the space X−s . We denote by X−∞ , X∞ the linear spaces X−∞ = ∪Xs , X∞ = ∩Xs . The space X∞ is dense in each Xs . It is formed by smooth (vector) functions. We shall often treat scales {Xs } as abstract Hilbert scales and do not discuss their embeddings to a scale of Sobolev functions. Example 1.1. Let Xs = H0s (S 1 , R) be the Sobolev space of 2πperiodic functions with zero meanvalue. This scale satisﬁes (i)–(iii). For a basis {φk  k ∈ Z0 } we take the trigonometric basis: 1 1 ϕk = √ cos kx, ϕ−k = − √ sin kx for k = 1, 2, . . . π π
(1.1)
(the minussign is introduced for further purposes). For a sequence ϑk we take ϑk = k. This choise corresponds to the homogeneous scalar product in Xs , u, vs = (u(s) v (s) )dx. Complexiﬁcation Xsc of this space is the space H0s (S 1 ; C) of complex Sobolev functions.
64
S. B. Kuksin
Given two scales {Xs }, {Ys } as above and a linear map L : X∞ → Y−∞ , we denote by Ls1 ,s2 ≤ ∞ its norm as a map Xs1 → Ys2 . We say that the map L deﬁnes a morphism of order d of the scales {Xs } and {Ys } for s ∈ [s0 , s1 ], if Ls,s−d < ∞ for each s ∈ [s0 , s1 ] with some ﬁxed −∞ ≤ s0 ≤ s1 ≤ +∞. 2 If in addition the inverse map L−1 exists and deﬁnes a morphism of order −d of the scales {Ys }, {Xs } for s ∈ [s0 + d, s1 + d], we say that L deﬁnes an isomorphism of order d of the two scales. If {Ys } = {Xs }, then an isomorphism L is called an automorphism. We shall drop the speciﬁcation “for s ∈ [s0 , s1 ]” if the segment [s0 , s1 ] is ﬁxed for a moment or can be easily recovered. If L : Xs → Ys−d is a morphism of order d for s ∈ [s0 , s1 ], then the adjoint maps L∗ : (Ys−d )∗ = Y−s+d → (Xs )∗ = X−s form a morphism of the scales {Ys } and {Xs } of the same order d for s ∈ [−s1 + d, −s0 + d]. We call it the adjoint morphism. A morphism L of a Hilbert scale {Xs }, complex or real, is called symmetric (anti symmetric) if L = L∗ (respectively L = −L∗ ) on the space X∞ . In particular, a linear operator L : Xs0 → Ys0 −d is called symmetric (anti symmetric) if L = L∗ (L = −L∗ ) on the space X∞ . If L is a symmetric morphism of {Xs } of order d for s ∈ [s0 , d − s0 ], then L∗ also is a morphism of order d for s ∈ [s0 , d − s0 ] and L = L∗ as the scale’s morphisms. We call such a morphism selfadjoint. Anti selfadjoint morphisms are deﬁned similar. Example. The operator − deﬁnes a selfadjoint automorphism of order 2 of the Sobolev scale {H0s }. The operator ∂/∂x deﬁnes an anti selfadjoint automorphism of order one. Linear maps from one Hilbert scale to another obey the Interpolation Theorem: Theorem 1 (see [LM, RS2). ] Let {Xs }, {Ys } be two real Hilbert scales and L : X∞ → Y−∞ be a linear map such that La1 ,b1 = C1 , La2 ,b2 = C2 . Then for any θ ∈ [0, 1] we have La,b ≤ Cθ , where a = aθ = θa1 + (1 − θ)a2 , b = bθ = θb1 + (1 − θ)b2 and Cθ = C1θ C21−θ . This result with Cθ replace by 4Cθ remain true for complex Hilbert scales. In particular, if under the theorem’s assumptions a1 − b1 = a2 − b2 =: d, then L deﬁnes a morphism of order d of the scales {Xs }, {Ys } for s ∈ [a1 , a2 ]. Corollary. Let L be a selfadjoint or an anti selfadjoint morphism of a scale {Xs } such that La,b = C < ∞ for some a, b. Then Lθ(a+b)−b,θ(a+b)−a ≤ C for any 0 ≤ θ ≤ 1. Proof. Since (Xa )∗ = X−a and (Xb )∗ = X−b , then C = La,b = L∗ −b,−a = L−b,−a . Now the assertion follows from the Interpolation Theorem. 2
if s0 = −∞, then s > s0 since X−∞ and Y−∞ are given no norms. Similar s < ∞ if s1 = ∞.
KAMpersistence of ﬁnitegap solutions
65
In particular, an operator L as above deﬁnes a morphism of order a − b of the scale {Xs } for s ∈ [−b, a] (or ∈ [a, −b] if −b > a). Let −∞ < a ≤ b ≤ ∞ and Os ⊂ Xs , s ∈ [a, b], be a system of compatible domains (i.e., Os1 ∩ Os2 = Os2 if s1 ≤ s2 ). Let F : Oa → Ya−d be an analytic (or C 1 smooth) map such that its restriction to the domains Os with a ≤ s ≤ b deﬁne analytic (or C 1 smooth) maps F : Os → Ys−d . 3 Then we say that F is an analytic (or C 1 smooth) map of order d for a ≤ s ≤ b. Example 1.1, continuation. Let us denote by Π the projector Π : H s → H0s , which sends a function u(x) to u(x) − u(x) dx/2π. The Sobolev spaces H s with s > 1/2 are Banach algebras: uvs ≤ Cs us vs , see [Ad]. Therefore for any segment [a, b], 1/2 < a ≤ b ≤ ∞, the map u(x) → Π◦F (u(x)) where F is a polynomial, deﬁnes an analytic map H0s → H0s of order zero for s ∈ [a, b]. If g(x) is any ﬁxed function, then the map u(x) → Π(F (u(x)) + g(x)) is analytic of order zero for s ∈ [a, b] if and only if g ∈ H b . The same is true for a map deﬁned by an analytic function F . More general, this is true for the map u(x) → Π(F (u(x), x)) where F (u, x) is a C b smooth function of u and x, which is δanalytic in u with some xindependent δ > 0. 1 Given a C smooth (or analytic) functional H : Xa ⊃ Oa → R , we identify the adjoint space (Xa )∗ with X−a and consider the gradient map ∇H : Oa → X−a ,
∇H(u), v = H∗ (u)v
∀v ∈ Xa .
If the domain Oa belongs to a system of compatible domains Os (a ≤ s ≤ b) and the gradient map ∇H is a C 1 smooth (or analytic) map of order dH in this system of domains, we write ord ∇H = dH . In this case the linearised gradient map ∇H(u)∗ : Xs → Xs−dH is bounded for any u ∈ Os with a ≤ s ≤ b. Since ∇H(u)∗ v1 , v2 = d2 H(u)(v1 , v2 ) = ∇H(u)∗ v2 , v1 , then the map ∇H(u)∗ is symmetric and due to the Corollary from Theorem 1 it deﬁnes bounded linear maps ∇H(u)∗ : Xs → Xs−dH for s ∈ [dH − d, d] (assuming that 2d ≥ dH ). 1.3
Diﬀerential forms
For d ≥ 0 and a domain O in a Hilbert space Xd from a Hilbert scale {Xs } we identify tangent spaces Tx O with Xd and treat diﬀerential kforms on O as continuous functions O × (Xd × · · · × Xd ) −→ R , which are polylinear k
and skewsymmetric in the last k arguments (see more in [Ca, La]). We write def
1forms as a(x) dx, where a : O → X−d and a(x) dx[ξ] →= a(x), ξ for ξ ∈ Xd . Besides, we write 2forms as A(x) dx ∧ dx, where def
A(x) dx ∧ dx[ξ, η] →= A(x)ξ, η0 3
for ξ, η ∈ Xd ,
if b = ∞, then a ≤ s < ∞. This agreement applies everywhere below.
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and A(x) : Xd → X−d is a bounded anti selfadjoint operator, i.e., its adjoint A(x)∗ : Xd → X−d equals −A(x). All the forms we consider below are analytic, where a kform ωk on O ⊂ Xd is called analytic if the corresponding map from O to the linear space of skewsymmetric polylinear functions (Xd × · · · × Xd ) −→ R is analytic. k
To deﬁne the diﬀerential of a C 1 smooth kform ωk we use the Cartan formula: d ωk (x)[ξ1 , . . . , ξk+1 ] =
k+1
(−1)i−1
i=1
∂ ωk (x)[ξ1 , . . . , ξˆi , . . . , ξk+1 ] . ∂ξi
(1.2)
Here the vectors ξj ∈ Tx O " Xd are extended to constant vector ﬁelds on O. So the r.h.s. of (1.2) is welldeﬁned and the commutatorterms from the r.h.s. of the classical Cartan formula (see e.g. [Go, La]) vanish. This deﬁnition well agree with the ﬁnitedimensional situation, as states the following obvious lemma: Lemma 1. Let ωk be a kform on a domain O ⊂ Xd , L be a ﬁnitedimensional aﬃne subspace of Xd and LO = L ∩ O. Then dωk LO = d(ω LO ). Proof. Both forms are given by the same formula (1.2).
Example 1.2. 1) The diﬀerential of a C 1 function f on O (=a zeroform) equals df = ∇f (x) dx. 2) The diﬀerential of a 1form a(x) dx, a : O → X−d , equals d(a(x) dx) = (a(x)∗ − a(x)∗ ) dx ∧ dx. Indeed, A(x) = a(x)∗ − the operator a(x)∗ : Xd → X−d is anti selfadjoint and d a(x)dx [ξ, η] = a(x)∗ ξ, η − a(x)∗ η, ξ = A(x)ξ, η. In the sequel we shall also work with kforms in subdomains of the direct products Zd , Zd = X × Yd , Zd z = (x, y), where X is a ﬁnitedimensional Euclidean space and Yd is a space from a Hilbert scale {Ys }.4 We write linear operators A in Zd in the blockform, AXX AXY , A= AY X AY Y where AXY : Yd → X, AY X : X → Yd and AXX : X → X, AY Y : Yd → Yd are bounded linear operators. The operator A is anti selfadjoint (with respect to the scalar product in X × Y0 ) if AXY = −AY∗ X and AXX , AY Y are anti selfadjoint operators. Accordingly we write the 2form A dz ∧ dz as A(z) dz ∧ dz = AXX (x, y) dx ∧ dx + AXY (x, y) dy ∧ dx + +AY X (x, y) dx ∧ dy + AY Y (x, y) dy ∧ dy. 4
Obviously, the spaces {Zs } also form a Hilbert scale.
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We note that in our notations AY X (x, y) dx∧dy[(δx1 , δy1 ), (δx1 , δy2 )] = AY X δx1 , δy2 Y = −δx1 , AXY δy2 . For subdomains of the manifolds Yd , Yd = Rn × Tn × Yd = {(p, q, y)},
(1.3)
we use natural versions of the notations given above. The Poincar`e lemma states that “locally” each closed form is exact. The proof is constructive and is well applicable to inﬁnitedimensional problems (see [Ca, La]). We shall need a version of the lemma for a closed 2form deﬁned in a neighbourhood O ⊂ Yd of the set P × Tn × {0}, where P is a subdomain of Rn , such that ﬁbres of the natural ﬁbration O → Rn × Tn are convex. Below we state the result, denoting by w points from Rn × Tn : Lemma 2. If ω2 (w, y) is a closed 2form in O and ω2 (w, 0) = 0, then ω2 = dω1 , where
1
ω1 (w, y)(δw, δy) =
ω(w, ty)[(0, y), (δw, tδy)] dt. 0
In particular, if ω2 = AW W (w, y)dw∧dw+AW Y (w, y) dy∧dw−A!∗W Y (w, y) dw 1 ∧ dy, then ω1 = a(w, y) dw, where a(w, y) = 0 AW Y (w, ty) dt y. This result follows from its ﬁnitedimensional version (see [AG]) and Lemma 1. Indeed, For any (w, y) ∈ O and ξ1 , ξ2 ∈ T(w,y) Yd " R2n × Yd = Zd we denote by Q a suﬃciently small neighbourhood of (w, y) in Yd and treat Q as a domain in Zd . Now we take for L the aﬃne 3space through (w, y) in the directions (0, y), ξ1 , ξ2 and get that dω1 (w, y)[ξ1 , ξ2 ] = ω2 (w, y)[ξ1 , ξ2 ].
2 2.1
Symplectic structures and Hamiltonian equations Basic deﬁnitions
¯ dx ∧ In a domain Od ∈ Xd with d ≥ 0 let us take a closed 2form α2 = J(x) ¯ dx such that the operator J analytically depends on x ∈ Od and deﬁnes a ¯ : Xd −→ ∼ Xd+dJ , dJ ≥ 0. Since the operator is anti linear isomorphism J(x) selfadjoint in X0 , then by the Corollary from Theorem 1 it deﬁnes an anti selfadjoint automorphism of order −dJ of the scale {Xs } for s ∈ [−d − dJ , d]. The form α2 supplies Od with a symplectic structure which corresponds to any C 1 smooth function h on Od the Hamiltonian vector ﬁeld Vh deﬁned by the usual (see [A1]) relation: α2 [Vh , ξ] = −dh(ξ) for all ξ ∈ T Od . For any ¯ x ∈ Od we have J(x)V h (x), ξ = −∇h(x), ξ for each ξ ∈ Xd . Thus, Vh (x) = J(x)∇h(x),
where
¯ −1 . J = (−J)
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The operator J(x) is called an operator of the Poisson structure. For each x it deﬁnes an anti selfadjoint automorphism of the scale of order dJ and the operator −→
J(x) : Xs+dJ ∼ Xs ,
−d − dJ ≤ s ≤ d,
(2.1)
analytically depend on x ∈ Od (see the Corollary to Theorem 1). Since the functional h is C 1 smooth, then the gradient map ∇h : Od → X−d is continuous. Using (2.1) we get that the vector ﬁeld Vh deﬁnes a continuous map Od → X−d−dJ . Usually we shall impose an additional restriction and assume that the vector ﬁeld Vh is smoother than that and ord Vh =: d1 < 2d + dJ . To stress that a domain Od ⊂ Xd is given a symplectic structure as above we shall write it as a pair (Od , α2 ). If the form α2 is analytic on the whole space Xs for each s ≥ s0 with some ﬁxed s0 , we shall say that ({Xs }, α2 ) is a symplectic Hilbert scale. A basis {φj (x)  j ∈ Zn } of a tangent space Tx Od = Xd is called symplectic if α2 [φj (x), φk (x)] = νj (x)δj,−k
(2.2)
for any j ∈ N and each k ∈ Z0 , with some positive real numbers νj (x), j ∈ N. For any C 1 smooth function h on Od × R a Hamiltonian equation with the hamiltonian h(x, t) is the equation x˙ (t) = J(x)∇h(x, t) =: Vh (x, t).
(2.3)
If ord Vh = 0 and the vector ﬁeld Vh is C 1 smooth and Lipschitz in Od , then the initialvalue problem for the equation (2.3) is wellposed: for any given initial condition x(0) ∈ Od it has a unique solution deﬁned while it stays in Od . A partial diﬀerential equation, supplemented by appropriate boundary conditions, is called a Hamiltonian PDE if under a suitable choice of a symplectic Hilbert scale ({Xs }, α2 ), a domain Od ⊂ Xd and a hamiltonian h, the equation can be written in the form (2.3). In this case the vector ﬁeld Vh is unbounded, ord Vh = d1 > 0: Vh : Od × R → Xd−d1 .
(2.4)
Usually the domain Od belongs to a system of compatible domains Os , s ≥ d0 , and the map Vh is analytic of order d1 for s ≥ d0 . For a vector ﬁeld Vh as in (2.4) with d1 > 0, diﬀerent classes of solutions for (2.3) can be considered. We choose the following deﬁnition: a continuous curve x : [0, T ] → Od is a solution of (2.3) in a space Xd if it deﬁnes a C 1 smooth map [0, T ] → Xd−d1 and both parts of (2.3) coincide as curves in Xd−d1 . A solution x(t) is called smooth if it deﬁnes a smooth curve in each space Xl . If a solution x(t), t ≥ τ , of (2.3) with x(τ ) = xτ exists and is unique, we write x(t) = Sτt xτ , or x(t) = S t−τ xτ if the equation is autonomous. The operators Sτt and S τ are called ﬂowmaps of the equation.
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For an equation (2.3) with d1 > 0 there is no general existence theorem for a solution of the corresponding initialvalue problem which would guarantee existence of the ﬂowmaps. To prove the existence is an art we do not touch here. Example 2.1 (semilinear equation). If (2.3) is a semilinear equation, i.e., Vh = A + V 0 , where A is an unbounded linear operator with a discrete imaginary spectrum and the nonlinearity V 0 is Lipschitz on bounded subsets of Xd , then solutions of the equation with prescribed initial conditions are well deﬁned till they stay in a bounded part of Xd , see [Paz]. Some important Hamiltonian PDEs are semilinear. For example, the nonlinear Schr¨ odinger equation: u(t, ˙ x) = i(∆u + f (u2 )u), x ∈ S 1 , where f is a smooth realvalued function (see [K, Paz]). Still, the semilinearity assumption is very restrictive since it fails for many important Hamiltonian PDEs (e.g., for the KdV). Our main concern are quasilinear Hamiltonian equations, i.e., equations (2.3) with Vh = A + V 0 , where A is a liner operator and ord A >ord V 0 . Possibly ord V 0 > 0 i.e., the equation may be nonsemilinear. Let Q ⊂ Od be a subdomain such that the ﬂowmaps maps Sτt : Q → Od are welldeﬁned and are C 1 smooth for τ ≤ t ≤ T , where −∞ ≤ τ < T ≤ ∞. Then diﬀerentiating a solution x(t) of (2.3) in the initial condition we get that the curve ζ(t) := Sτt x(τ ) ∗ ζ satisﬁes the linearised equation ˙ = Vh (x(t), t)∗ ζ(t), ζ(t)
ζ(τ ) = ζ.
(2.5)
The assumption that the map Sτt is C 1 smooth in a subdomain is very restrictive since to check the smoothness of ﬂowmaps for many important equations (even for the KdV!) is a nontrivial task. To get rid of it we give the following Deﬁnition 1. Let x(t), t ∈ R, be a solution for equation (2.3). If for each ζ ∈ ˙ = Vh (x(t), t)∗ ζ(t), ζ(θ) = ζ, has Xd and each θ the linearised equation ζ(t) a unique solution ζ(t) ∈ Xd deﬁned for all t and such that ζ(t)d ≤ Cζd t (x)ζ and uniformly in θ, t from a compact segment, then we write ζ(t) = Sθ∗ t say that ﬂow {Sθ∗ (x)} of the linearised equation (2.5) is well deﬁned in Xd . The property described in Deﬁnition 1 characterises the ﬂow only in the “inﬁnitesimal vicinity” of a solution of (2.3). It suits well our goal to study special families of solutions rather than the whole ﬂow of the equation. If the ﬂowmaps Sτt are C 1 smooth, then Sτt (x)∗ = Sτt ∗ (x), but the map in the r.h.s. of this relation can be well deﬁned while the map in the l.h.s. is not. Example 2.2 (Equations of the Korteweg–de Vries type). Let us take for {Xs } the scale of Sobolev spaces H0s as in Example 1.1. We deﬁne a ∂ Poisson structure by means of the operator J = ∂x , so dJ = 1 and −J¯
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S. B. Kuksin
is the operator (∂/∂ x)−1 of integrating with zero meanvalue. We get the symplectic Hilbert scale ({H0s }, −(∂/∂ x)−1 du ∧ du). We stick to the discrete scale {s ∈ Z}: it is suﬃcient since the orders of all involved operators are integer. The trigonometric basis {ϕj  j ∈ Z0 } (see (1.1)) is symplectic since for j ≥ 1 and any k we have: ¯ −1/2 cos jx, ϕk (x) = j −1 −π −1/2 sin jx, ϕk (x) = j −1 δj,−k . α2 [ϕj , ϕk ] = Jπ 2π For a hamiltonian h we take h(u) = 0 (− 18 u (x)2 + f (u)) dx with some analytic function f (u). Then ∇h(u) = 14 u + f (u) 5 and Vh (u) = 14 u + ∂ ∂x f (u). Thus the Hamiltonian equation takes the form u(t, ˙ x) =
1 ∂ u + f (u) 4 ∂x
(2.6)
1 3 The maps H0s → R, u(x) → (for f (u) = 4 u swe get sthe KdV equation). f (u) dx and H0 → H , u(x) → f (u(x)), with s ≥ 1 are analytic (see Example 1.1). So the map H0s u → Vh (u) ∈ H0s−3 is analytic for s ≥ 1. That is, the vector ﬁeld Vh is analytic of order 3 for s ≥ 3. Being supplemented by an initial condition u(0, x) = u0 (x) ∈ H0s with s ≥ 3, equation (2.6) has a unique solution in H0s . This solution exists for t < T (u0 s ) (T is a continuous positive function) and the ﬂowmaps S t , t < T , are C 1 smooth. This is a nontrivial result, see e.g. [Kat1]. On the contrary, if u(t, x) is a smooth solution of (2.6), then the linearised equation
v˙ =
∂ 1 v + (f (u)v), 4 ∂x
v(0, x) = v0 ∈ H0s ,
(2.7)
has a unique solution in H0s with any s ≥ 0 by trivial arguments, see [KK, Paz]. We shall often work with equations in a subdomain Od of the manifold Yd (d ≥ 0) as in (1.3), given a symplectic structure by means of a 2form (dp ∧ ¯ dq) ⊕ (Υ(y)dy ∧ dy), where dp ∧ dq is the classical symplectic form on Rn × T n ¯ and Υ(y)dy∧dy is a closed 2form in a domain in Yd . This symplectic structure corresponds to a C 1 smooth function H(p, q, y) the following Hamiltonian system: p˙ = −∇q H, q˙ = ∇p H, y˙ = Υ∇y H. Solutions of these equations are deﬁned in the same way as solutions of (2.3). 2.2
Symplectic transformations
Let {Xs }, {Ys } be two Hilbert scales and d, d˜ ≥ 0. Let O ⊂ Xd and Q ⊂ Yd˜ ¯ be two domains given symplectic structures by 2forms α2 = J(x)dx ∧ dx and 5
since dh(u)v =
− 14 u (x)v (x) + f (u(x))v(x) dx = 14 u (x) + f (u(x)), v(x)L2 .
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71
¯ β2 = Υ(y)dy ∧ dy as in Section 2.1. A C 1 smooth map Φ : Q → O is called symplectic if Φ∗ α2 = β2 . That is, if for any y ∈ Q with Φ(y) = x ∈ O we have ¯ ¯ J(x)Φ ∗ (y)ξ, Φ∗ (y)ηX0 = Υ(y)ξ, ηY0 for all ξ, η ∈ Yd˜, or ¯ ◦ Φ∗ (y) = Υ(y). ¯ Φ∗ (y) ◦ J(x)
(2.8)
A symplectic map Φ is an immersion since by (2.8) its tangent maps are embeddings. If a symplectic map Φ is such that the tangent maps Φ∗ (y) deﬁne isomorphisms of the spaces Yd˜ and Xd , then Φ is called a symplectomorphism. We shall need an obvious version of the deﬁnitions above for the case when Oc and Qc are complex domains in Xdc and Yd˜c respectively and the forms α2 , β2 have constant coeﬃcients: an analytic map Φ1 : (Qc , α2 ) → (Oc , β2 ) ¯ 1 (y)∗ ξ, Φ1 (y)∗ ηX ≡ Υξ, ¯ ηY . In particular, an analytic is symplectic if JΦ 0 0 symplectomorphism Φ : Q → O analytically extends to a symplectomorphism of suﬃciently small complex neighbourhoods of Q and O (the forms α2 , β2 are assumed to be constantcoeﬃcient). From now on for the sake of simplicity we restrict ourselves to the case we need below: d = d˜ ≥ 0,
¯ = ord Υ(y) ¯ ord J(x) = −dJ
∀ x, y.
¯ = J¯ and Υ(y) ¯ ¯ are constants Proposition 1. Let us assume that J(x) =Υ isomorphisms of the corresponding scales of order −dJ . Let Φ : (Qc , β2 ) → (Oc , α2 ) be a symplectomorphism such that Φ(y)∗ d,d , (Φ(y)∗ )−1 d,d ≤ C for every y ∈ Qc . Then Φ(y)∗ θ,θ , (Φ(y)∗ )−1 θ,θ ≤ C1 for every y ∈ Qc and every θ ∈ [−d − dJ , d]. If Φ is an analytic symplectomorphism, then these maps are analytic in y ∈ Qc . ¯ ∗ ◦J. So Φ(y)∗ d+d ,d+d ≤ C for every Proof. By (2.8) we have Φ∗ = −Υ◦Φ J J y. Hence, Φ(y)∗ −d−dJ ,−d−dJ ≤ C and the estimate for Φ∗ θ,θ follows by interpolation. The estimates !for Φ−1 follow from the identity (Φ∗ )−1 = ∗ ∗ −J¯◦Φ∗ ◦Υ since Φ−1 = (Φ∗ )−1 = (−J¯◦Φ∗ ◦Υ)∗ is a zeroorder morphism ∗
for s ∈ [−d − dJ , d]. In the analytic case the maps are analytic in y by the criterion of analyticity. As in the ﬁnitedimensional case, symplectic maps transform Hamiltonian equations to Hamiltonian. Let Φ : Q → Oc be an analytic symplectic map such that Φ(y)∗ : Ys → Xs is a linear map, analytic in y ∈ Q, for any s ≤ d. (2.9) ¯ ¯ ¯ are constant isomorphisms of zero We note that if J(x) = J¯ and Υ(y) =Υ order, then the assumption (2.9) is satisﬁed due to Proposition 1. Theorem 2. Let domains O ⊂ Xd and Q ⊂ Yd , d ≥ 0, be given symplectic ¯ = −dJ . structures by analytic 2forms α2 , β2 as above with ord J¯ =ord Υ
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Let the vector ﬁeld Vh = J∇h of equation (2.3) deﬁnes a C 1 smooth map Vh : O × R → Xd−d1 of order d1 ≤ 2d and let Φ : Q → O be an analytic symplectic map satisfying (2.9), such that the vector ﬁeld Vh in O is tangent to Φ(Q) in the following sense: Vh (Φ(y)) = Φ(y)∗ ξ
for any y ∈ Q with an appropriate ξ ∈ Yd−d1 . (2.10)
y˙ = Υ(y)∇y H(y, t),
¯ −1 , H = h ◦ Φ , Υ = (−Υ)
(2.11)
to solutions of (2.3). We note right away that the assumption (2.10) becomes empty if Φ is a symplectomorphism (to be more speciﬁc, now (2.10) follows from (2.9) since d − d1 ≥ −d). Proof. Let y(t) be a solution of (2.11). By (2.9) the curve x(t) = Φ(y(t)) is C 1 smooth in Yd−d1 and is continuous in Yd . It remains to check that it satisﬁes (2.3). Since x˙ = Φ(y)∗ y˙ and ∇y H = Φ(y)∗ ∇x h, then ¯ x˙ = Φ(y)∗ Υ(y)Φ(y)∗ ∇x h = −Φ(y)∗ Υ(y)Φ(y)∗ J(x)V h (x). ¯ By (2.10), Vh (x) = Φ(y)∗ ξ. So the r.h.s is −Φ(y)∗ Υ(y)Φ(y)∗ J(x)Φ(y) ∗ ξ. By ¯ (2.8) it equals −Φ(y)∗ Υ(x)Υ(x)ξ = Φ(y)∗ ξ = Vh (x). Thus, x(t) satisﬁes the equation (2.3). To apply Theorem 2 we have to be able to construct suﬃcient amount of symplectic transformations. An important way to construct symplectomorphisms of subdomains of (O ⊂ Xd , α2 ) is to get them as ﬂowmaps Stτ of an additional nonautonomous Hamiltonian equation x˙ = J(x)∇x f (t, x) = Vf (t, x), where the hamiltonian f is such that the vector ﬁeld Vf is Lipschitz, see [K, KK]. ¯ dx ∧ dx). Let O be a domain in a symplectic space (Xd , α2 = J(x) Deﬁnition 2. Let C 1 smooth functions H1 , H2 on a domain O ⊂ Xd deﬁne continuous gradient maps of orders d1 , d2 ≤ 2d such that d1 + d2 + dJ ≤ 2d. Then the Poisson bracket {H1 , H2 } of H1 , H2 is the continuous on O function {H1 , H2 }(x) = J(x)∇H1 (x), ∇H2 (x). The scalar product J(x)∇H1 , ∇H2 is welldeﬁned and is continuous in x. The Poisson bracket is skewsymmetric, {H1 , H2 } = −{H2 , H1 }. In particular, {H, H} ≡ 0 (if ord ∇H ≤ d − dJ /2). 2.3
Darboux lemmas
The classical Darboux lemma states that locally near a point any closed nondegenerate 2form in R2n can be written as dp ∧ dq. This result has several
KAMpersistence of ﬁnitegap solutions
73
versions which put a closed nondegenerate 2form on a manifold to diﬀerent normal forms in the vicinity of a closed set (for the classical lemma the set is a point), see [AG]. Some of these results admit direct inﬁnitedimensional reformulations which can be proven by the same arguments due to Moser– Weinstein. In this section we present a version of the Darboux lemma which will be used later on. Let Yd = Rn × T n × Yd and W be its subset of the form W = P × n T × {0}, where P is a bounded subdomain of Rn . By O, O1 , . . . we denote δneighbourhoods of W in Yd with diﬀerent δ > 0 and suppose that in a neighbourhood O we are given two closed analytic 2forms ω0 and ω1 , both of them of the form ωj = J¯j (y) dy ∧ dy, where y = (p, q, y) and dy = (dp, dq, dy). We assume that (i) ω0 = ω1 in T O W , (ii) for all t ∈ [0, 1] and all y ∈ O the map J¯t (y) = (1 − t)J¯0 + tJ¯1 deﬁnes an isomorphism J¯t : Zd →−→ Zd+dJ with some ﬁxed dJ ≥ 0, where ∼ Zd = Rn × Rn × Yd . By (ii), the map J t = (−J¯t )−1 : Zd+d →−→ Zd is well deﬁned and J
∼
analytically depends on y. By Poincar´e’s lemma (see Lemma 2 above), the form ω1 − ω0 equals dα for some analytic oneform α = a(y) dy such that a(p, q, y) = O(y2d ). Let us also assume that (iii) the map O1 → Zd+dJ , y → a, is Lipschitz analytic in a neighbourhood O1 of W . Lemma 3 (Moser–Weinstein). Under the assumptions i)iii) there exists a neighbourhood O2 and an analytic diﬀeomorphism ϕ : O2 → O such that ϕ W = id, ϕ∗ W = id and ϕ∗ ω1 = ω0 . Moreover, ϕ equals to a timeone ﬂowmap S01 , corresponding to the nonautonomous equation y˙ = J t a(y) =: V (t, y). Proof. For 0 ≤ t ≤ 1 let us consider the 2forms ωt = (1 − t)ω0 + tω1 = J¯t dy ∧ dy. These forms are closed as well as the forms ω0 , ω1 and are nondegenerate in a neighbourhood O3 since ωt = ω0 = ω1 on W by i). Now we denote by ϕt the ﬂowmaps S0t of equation y˙ = V (t, y); so ϕ0 = id, ϕ1 = ϕ and (ϕ1 − id)(p, q, y) = O(y2d ). The lemma will be proven if we check that (ϕt )∗ ωt = const. Because Cartan’s identity (see e.g. [GS] and see [KK] for t the inﬁnitedimensional case) we have to prove that ∂ω ∂t + d(V %ωt ) = 0. Since ∂ωt /∂t = ω1 − ω0 = dα, then it remains to check that α + V %ωt ≡ 0. But V %ωt = V %J¯t dy ∧ dy = (J¯t V )dy. So α + V %ωt = (a + J¯t V ) dy = (a + J¯t J t a) dy = 0 and the lemma is proven. Appendix. Timequasiperiodic solutions Our main goal is to study timequasiperiodic solutions x(t) of some Hamiltonian equations (2.3). Here we recall corresponding basic deﬁnitions.
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Deﬁnition. A C 1 curve γ : R → X in a Banach space or a manifold X is called quasiperiodic (QP) with ≤ n frequencies if there exists a C 1 smooth map Γ : T n → X, a vector ω ∈ Rn and a point q0 ∈ T n such that γ(t) ≡ Γ(q0 + ωt) .
(A.1)
The vector ω is called the frequency vector and q0 is called the phase. The minimal n such that γ(t) admits a representation (A.1) is called the number of independent frequencies; corresponding numbers ω1 , . . . , ωn are called the basic frequencies and the numbers 2π/ω1 , . . . , 2π/ωn – the basic periods. Remark. We note that the vector ω formed by the basic frequencies is deﬁned only up to an unimodular transformation L since the curve γ(t) can be also written as γ(t) = ΓL (Lq0 + Lωt), where ΓL (q) = Γ(L−1 q). What is uniquely deﬁned, is the Zmodule Z ω1 + Z ω2 + · · · + Z ωn ⊂ R. We shall usually ignore this subtlety. The closure γ(R) is called the hull of γ. If components of ω are rationally independent, then the hull equals Γ(Tn ). We call a solution x of (2.3) a (time) quasiperiodic, or a QP solution, if the curve x : R → Xd is QP, and call it analytic quasiperiodic if the corresponding map Γ is analytic of maximal rank. The hull Γ(T n ) of an analytic QP solution (A.1) with n basic frequencies is an invariant analytic ntorus of the equation. This torus is an analytic submanifold of X if the map Γ is an immersion.
3
Laxintegrable Hamiltonian equations and their integrable subsystems
We consider a symplectic Sobolev scale ({Zs }, α2 ), α2 = J¯ dz ∧ dz, as above. The operator J¯ deﬁnes an anti selfadjoint automorphism of the scale of a negative order −dJ ≤ 0. To a hamiltonian H = 12 Az, z + H(z), where A is a selfadjoint morphism of the scale of order dA , the symplectic structure corresponds the Hamiltonian equation u˙ = J∇H(u) = J(Au + ∇H(u)) =: VH (u),
¯ −1 . J = (−J)
(3.1)
We assume that the hamiltonian H is analytic quasilinear, that is, the functional H is analytic on a domain Od ⊂ Zd , d ≥ dA /2, and deﬁnes an analytic gradient map of order dH < dA , ∇H : Od → Zd−dH . By this assumption and the Corollary to Theorem 1, for any u ∈ Od the linear map ∇H(u)∗ deﬁnes a morphism of the scale {Zs } of order dH for s ∈ [−d + dH , d]. We shall study equation (3.1) in a suﬃciently smooth space Zd taking any d such that d1 := ord VH = dA + dJ ≤ 2d + dJ . We do not assume that the ﬂow maps of the equation are deﬁned on the whole domain Od (i.e., we do not assume that the equation can be solved for any initial condition u(0) ∈ Od ).
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75
Examples of Hamiltonian PDEs
Quasilinear Hamiltonian PDEs with analytic coeﬃcients have the form (3.1), where usually Od equals to the whole space Zd and the gradient map ∇H is analytic of some order dH for all suﬃciently smooth spaces Zd (i.e., it is an analytic map of order dH for d ≥ d0 with some ﬁxed d0 ). The following examples and their perturbations will be the main through our work: Example 3.1 (KdV equation, cf. Example 2.2). Let us take for a scale {Zs  s ∈ Z} the scale {H0s (S 1 ; R)} of 2πperiodic Sobolev functions with zero mean value, deﬁned in Example 1.1. We choose! J = ∂/∂x, A = 14 ∂ 2 /∂x2 and 2 − 18 u + 14 u3 dx. The equation (3.1) takes H(u) = 14 u3 dx, so H(u) = the form: u˙ =
1 ∂ (uxx + 3u2 ). 4 ∂x
(KdV)
It is considered under zero meanvalue periodic boundary conditions: u(t, x) ≡ u(t, x + 2π),
2π
u(t, x) dx ≡ 0, 0
which are satisﬁed automatically since we are looking for solutions in a space H0s . The gradient map Zd → Zd , u → ∇H = 34 u2 , is analytic of order dH = 0 for d ≥ 1 (see in Example 2.2). Now we have ord A = dA = 2 and ord J = dJ = 1. Example 3.2 (higher KdV equations). The KdV equation in the previous example is an equation from an inﬁnite hierarchy of Hamiltonian equations, called the KdVhierarchy [DMN, McT, ZM]. The lth equation from the hierarchy can be written as an equation (3.1) in the same symplectic Hilbert scale ({H0s }, J du, du). It has a hamiltonian Hl of the form Hl (u) = 2π ! 2 Kl (−1)l u(l) + higherordertermswith ≤ l − 1 derivatives dx, 0
where Kl is a nonzero constant (H1 is just the KdVhamiltonian). In particular, the hamiltonian H2 has the form H2 = 18 (u2xx − 5u2 uxx − 5u4 ) dx and the corresponding Hamiltonian equation is the ﬁfth order partial diﬀerential equation: 1 ∂ 1 u˙ = u(5) − (5u2x + 5uuxx + 10u3 ). 4 4 ∂x The gradient map of the nonquadratic part of this hamiltonian deﬁnes in the Sobolev scale {H0s } an analytic map of order dH = 2 for any s ≥ 2. The order dA of the linear part equals 4 and dJ = 1.
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Example 3.3 (SineGordon equation). The SineGordon (SG) equation on the circle, u ¨ = uxx (t, x) − sin u(t, x),
x ∈ S = R/2πZ,
(SG)
corresponds to a bounded nonlinearity. For any u0 ∈ H s (S) and u1 ∈ H s−1 (S) this equation has a unique solution u(t, x) ∈ C(R, H s ) ∩ C 1 (R, H s−1 ) such that u(0, x) = u0 and u(0, ˙ x) = u1 . This is almost obvious, see [Paz]. The equation (SG) can be written in a Hamiltonian form in many diﬀerent ways. The most strightforward way is to write (SG) as u˙ = −v,
v˙ = −uxx + sin u(t, x).
(3.2)
To are Hamiltonian, we take the symplectic scale see thats these equations ¯ dξ , where ξ = (u, v) ∈ Zs and {Zs = H (S) × H s (S)}, α2 = Jdξ, J(u, v) = (−v, u) (so J¯ = J). For a hamiltonian H we choose H = 12 Aξ, ξ + H(ξ), where A(u, v) = (−uxx , v) and H(u, v) = − cos u(x) dx. Then ∇H(u, v) = (sin u, 0) and the Hamiltonian equation ξ˙ = J∇H = J(Aξ + ∇H(ξ)) coinsides with (3.2). The Hamiltonian form (3.2) is traditional (cf. [McK]) and is convenient to study explicit (“ﬁnitegap”) solutions of (SG), but not to carry out detailed analysis of the equation since the linear operator A as above deﬁnes a selfadjoint morphism of the scale {Zs } of order two, which is not an ordertwo automorphism (the invererse map A−1 deﬁnes a morphism of order 0, not −2). To get a Hamiltonian form of the SG equation, convenient for its analysis, we denote w = A−1/2 v, where A = −∂ 2 /∂x2 + 1, and write (SG) as √ √ u˙ = − A w, w˙ = A u + A−1 (sin u − u) . (3.3) The equations (3.3) have more symmetric form than (3.2). They turn out to be Hamiltonian in the shifted Sobolev scale {Zs = H s+1 (S) × H s+1 (S)}, see [K, BiK1] and [KK]. Even periodic boundary conditions. If u(t, x) is any solution of (SG) such that the initial conditions (u(0, x), u(0, ˙ x)) = (u0 (x), u1 (x)) are even periodic functions, i.e., u0 (x) ≡ u0 (x + 2π) ≡ u0 (−x)
(EP)
and similar with u1 , then u− (t, x) = u(t, −x) is another 2πperiodic solution for SG with the same initial conditions. Since a solution of the initialvalue problem for (SG) is unique, then u− (t, x) ≡ u(t, x). That is, the space of even periodic functions is invariant under the SGﬂow and we can study the equation under the boundary conditions (EP). These conditions clearly imply Neuman boundary conditions on the halfperiod: (N) u0x , u1x (0) = u0x , u1x (π) = (0, 0).
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The former can be viewed as a “smoother version” of the latter since for any smooth even periodic function all its oddorder deviatives (not only the ﬁrst one) coinside at x = 0 and x = π. Denoting for any real s by Zse a subspace of Zs , formed by even functions, we observe that the equation (SG)+(EP) can be written in the Hamiltonian ¯ dξ , where Z e = {ξ(x) ∈ form (3.2) in the symplectic scale {Zse }, Jdξ, s Zs  ξ satisﬁes (EP). We note that for s = 2 the space Z2e is formed by the vectorfunctions from H 2 (0, π) × H 2 (0, π) which satisfy (N) (the functions are assumed to be extended to the segment [0, 2π] in the even way). That is, for solutions of the SG equation in the Sobolev space H 2 , the boundary conditions (OP) and (N) are equivalent. Odd periodic boundary conditions. Similarly, the SGequation under the odd periodic boundary conditions u(t, x) ≡ u(t, x + 2π) ≡ −u(t, −x)
(OP)
can be written in a Hamiltonian form in the symplectic scale {Zso }, β2 = ¯ dξ , where Z o = {ξ(x) ∈ Zs  ξ satisﬁes (OP) }. These boundary Jdξ, s conditions imply the Dirichlet ones: u0 , u1 (0) = u0 , u1 (π) = (0, 0). 3.2
Laxintegrable equations
Let us consider a quasilinear Hamiltonian PDE and write it in the Hamiltonian form (3.1). This equation is called Laxintegrable if there exist linear operators Lu , Au which depend on u ∈ Z∞ and deﬁne linear morphisms of ﬁnite orders of some additional real or complex Hilbert scale {Zs }, such that u(t) is a smooth solution of (3.1) if and only if d Lu(t) = [Au(t) , Lu(t) ]. dt
(3.4)
The operators Lu and Au are said to form an LA pair of the equation (3.1)=(3.4). We specify dependence of the L, Aoperators on u and assume existence of integers s and d such that for all s ≥ s the maps u → Lu and u → Au are analytic as maps from Zs to the space of bounded linear operators Zd → Zd−d , provided that d ≤ s. Due to this assumption, the l.h.s of (3.4) is well deﬁned for any C 1 smooth curve u(t) ∈ Zs , s ≥ s . Both KdV and SineGordon equations are Laxintegrable, see below Section 4. We abbreviate Lt = Lu(t) and At = Au(t) , where u(t) is a smooth solution for (3.4). A crucial property of the L, Aoperators is that spectrum of the operator Lt is timeindependent and its eigenvectors are preserved by the ﬂow, deﬁned by the operators At :
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Lemma 4. Let χ0 ∈ Z∞ be a smooth eigenvector of L0 , L0 χ0 = λχ0 . Let us also assume that the initialvalue problem χ˙ = At χ,
χ(0) = χ0 ,
(3.5)
has a unique smooth solution χ(t) ∈ Z∞ . Then Lt χ(t) = λχ(t) for any t. Proof. Let us denote Lt χ(t) = ξ(t), λχ(t) = η(t) and calculate derivatives of these functions. We have: d d ξ = Lχ = [A, L]χ + LAχ = ALχ = Aξ dt dt d d and dt η = dt λχ = λAχ = Aη . Thus, both ξ(t) and η(t) solve the problem (3.5) with χ0 replaces by λχ0 and coincide for all t by the uniqueness assumption.
In many important examples of Laxintegrable equations, {Zs } is the Sobolev scale of Lperiodic in x (vector) functions and L, A are udependent diﬀerential operators, acting on complex vectorfunctions. In this case it is natural to take for the scale {Zs } the Sobolev scale of Lperiodic complex vectorfunctions. So Lperiodic spectrum of the operator Lu is an integral of motion for the equation (2.8) if the linear equation (2.11) deﬁnes a ﬂow in the space of smooth Lperiodic vectorfunctions. The set of integrals which we obtain in this way usually is incomplete. To get missing integrals we note that an Lperiodic in x solution u(t, x) can be also treated as an Lmperiodic solution for any m ∈ N. Accordingly, we can consider the L, Aoperators under mLperiodic boundary conditions and take for {Zs } the Sobolev scale of mLperiodic vectorfunctions. Due to the lemma, the mLperiodic spectrum of L is an integral of motion if the equation (2.11) deﬁnes a ﬂow in the corresponding space Z∞ . This set of integrals contains the initial one. 3.3
Integrable subsystems
Now we suppose that equation (3.1) possesses an invariant submanifold T 2n ⊂ Od ∩ Z∞ , such that restriction of the equation to T 2n is integrable. For some important examples the manifold T 2n may have singularities and the restricted symplectic form α2 T 2n may degenerate. Since our objects are analytic, these degenerations can only happen on singular subsets of positive codimension and do not aﬀect the ﬁnal KAMresults which neglect subsets of small measure: at some point we shall cut out the singular subsets with their small neighbourhoods. But our preliminary arguments are global . To carry them out we have to develop global notations. We assume that T 2n = Φ0 (R × T n ) where T n = {z} is the standard ntorus and R is a connected ndimensional analytic set which is the real part of a connected complex analytic subset Rc of a domain Πc ⊂ CN . 6 By Rsc
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we denote any proper analytic subset of Rc which contains its singular part and denote by Rs the real part of Rsc , i.e., Rs = Rsc ∩ R. We assume that the map Φ0 is analytic and the form α2 T 2n does not degenerate identically: The map Φ0 : R × T n → Zl is analytic for each l. That is, for some δ = δl > 0 it extends to an analytic map Πc × {Im z < δ} → Zlc . c such that the analytic 2form (ii) Rc contains a proper analytic subset Rs1 ∗ c Φ0 α2 is nondegenerate in (R \ (Rs ∪ Rs1 )) × T n , where Rs1 = Rs1 ∩ R. (i)
For brevity we redenote Rs := Rs ∪ Rs1 and similar redenote Rsc . We set R0c = Rc \ Rsc and R0 = R \ Rs . Since Rs and Rsc comprise singularities of the analytic sets R and Rc as well as of the map Φ0 , then the sets R0 and R0c are smooth analytic manifolds and the map Φ0 : R0 × T n → Zl , Φ0 (R0 × T n ) =: T02n , is an immersion. We assume that (iii) The set T02n is a smooth analytic submanifold of each space Zl , invariant for the equation (3.1), as well as the tori T n (r) = Φ0 ({r} × T n ), r ∈ R0 . The restricted to T n (r) equation takes the form z˙ = ω(r), where ω extends to an analytic map Πc → Cn . Due to (ii) and (iii), the manifold T02n is ﬁlled with smooth timequasiperiodic solutions of the equation (3.1). The frequency map r → ω(r) is assumed to be nondegenerate: (iv) for almost all r ∈ R0 , the tangent map ω∗ (r) : Tr R0 → Rn is an isomorphism. By Theorem 2 the equation restricted to T02n is Hamiltonian. Condition iv) implies that this equation is integrable (see [KK] or [BoK2], p. 106). So by the Liouville–Arnold theorem, R0 can be covered by a countable system of open domains R0j , R0 = R01 ∪ R02 ∪ · · · , such that the equation (3.1) restricted to each manifold Tj2n = Φ0 (R0j × T n ) admits actionangle variables (p, q) with actions p ∈ Pj Rn and angles q ∈ T n . I.e., ω2 = dp ∧ dq and the equation restricted to Tj2n takes the form p˙ = 0, q˙ = ∇h(p),
h = H ◦ Φ0 ,
where h = hj . Lemma 5. In any domain R0j we have ∇h(p(r)) = ω(r) and q = z + q 0 (r). For many important examples the set of integrals of motion of a Laxintegrable equation, formed by 2πperiodic and 2πantiperiodic spectra (see Lemma 4 and its discussion), is “complete”. Fixing all but some n of them one can construct invariant manifolds T 2n as above. Below we show how to carry out this construction for KdV and SG equations. 6
That is, Rc is formed by zeroes of an analytic map Πc → CN −n such that at some points of Πc its linearisation has full rank. For elementary facts concerning analytic sets, real and complex, see [Mil] and [GR], Sections II, III.
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S. B. Kuksin
Finitegap manifolds and thetaformulas
We start with famous ﬁnitegap solutions for the KdV equation under zeromeanvalue periodic boundary conditions: 2π 1 ∂ 2 (uxx + 3u ), u(t, x) ≡ u(t, x + 2π), u dx ≡ 0. (KdV) u˙ = 4 ∂x 0 The ﬁnitegap solutions ﬁll invariant submanifolds T 2n ⊂ H0s (S 1 ) with integrable dynamics on them, as in Section 2. To study the manifolds T 2n we shall use the Its  Matveev formula which represents the ﬁnitegap solutions in terms of thetafunctions. 4.1
Finitegap manifolds for the KdV equation 2
∂ The L, Aoperators for the KdV equation are Lu = − ∂x 2 − u and Au = ∂3 3 ∂ 3 + u + u . Indeed, calculating the commutator [A, L]v one sees that ∂x3 2 ∂x 4 x most of the terms cancel and there is nothing left except ( 14 uxxx + 32 uux )v. Thus, [A, L] is an operator of multiplication by the r.h.s. of KdV and the equation can be written in the form (3.4). For the scale {Zs } we take one of the following scales of complex functions: or the scale of 2πperiodic functions, or the scale of 2πantiperiodic functions, or the scale of 4πperiodic ones. It is wellknown [Ma, MT] that the spectrum of the Sturm–Liouville operator Lu acting on twice diﬀerentiable functions of period 4π is a sequence of simple or double eigenvalues {λj  j ≥ 0}, tending to inﬁnity:
λ0 < λ1 ≤ λ2 < λ3 ≤ λ4 < · · · & ∞. Corresponding eigenfunctions are smooth if the potential u(x) is. The spectrum {λj } can be also described without doubling the period: it equals the union of the periodic and antiperiodic spectra of the operator Lu , considered on the segment [0, 2π]. Below we denote λ = {λ0 , λ1 , . . . } and refer to the sequence λ = λ(u) as to the periodic/antiperiodic spectrum of the operator Lu . The segment ∆j = [λ2j−1 , λ2j ], j = 1, 2, . . . , is called the j th spectral gap. The gap ∆j is open if λ2j > λ2j−1 and is closed if λ2j = λ2j−1 . Example. For u = 0 we have λ2k = k 2 /4, k ≥ 0, and λ2l−1 = l2 /4, l ≥ 1. Corresponding eigenfunctions are (2π)−1/2 cos kx/2 and (2π)−1/2 sin lx/2. All gaps ∆j are now closed. If u(t, x) is a smooth xperiodic function, then the linear equation v˙ = Au(t,x) v, v(0, x) = v0 (x), has a unique smooth periodic solution v(t, x) for any given smooth periodic initial data v0 (x) (this follows from an abstract theorem in [Paz]). Hence, Lemma 4 with {Zs = H s (R/4πZ)} imply that the sequence λ is an integral of motion: λ(u(t, ·)) is timeindependent if u(t, x) is a solution of the KdV.
(4.1)
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Let us ﬁx any integer nvector Υ = (V1 , . . . , Vn ) ∈ Nn , where V1 < · · · < Vn , and consider a set TΥ2n , TΥ2n = {u(x)  the gap ∆j (u) is open iﬀ j ∈ {V1 , . . . , Vn }}. n This set equals to the union of isospectral " subsets T n (r) = TΥ (r) with 2n n n prescribed lengths of the open gaps: TΥ = r∈Rn TΥ (r), where TΥ (r) = +
n (r) is invariant for the {u(x) ∈ TΥ2n  ∆Vj  = rj ∀ j}. By (4.1) each set TΥ KdVﬂow. Remarkably, the whole spectrum λ of an ngap potential is deﬁned by n the nvector r and analytically depends on it [GT]. Each set TΥ (r) is not n empty and is an analytic ntorus in any space H0s = H0s (S 1 ). The tori TΥ (r) are analytically glued together, so TΥ2n is an analytic submanifold of each space H0s . – These are wellknown results from the inverse spectral theory of the SturmLiouville operator Lu , see [Ma] and [Mo, GT, MT]. So the inverse spectral theory provides us with KdVinvariant 2nmanifolds foliated to invariant ntori.
4.2
The Its–Matveev thetaformulas
To check that the ngap manifolds TΥ2n of the KdV equation possesses the properties (i)–(iv) from Section 2, we have to present an analytic map Φ0 as in Section 2 and to check its properties. We shall write the map Φ0 in terms of thetafunctions, following [D, BB]. n Let us take any ngap potential u(x) ∈ TΥ (r) and denote by E1 (r) < E2 (r) < · · · < E2n+1 end points of the open gaps plus λ0 (so E1 = λ0 and ∆V1 = [E2 , E3 ], . . . , ∆Vn = [E2n , E2n+1 ]). The Riemann surface Γ = Γ(r) of genus n, Γ = {P = (λ, µ)  µ2 = R(λ; r) :=
2n+1
(λ − Ej (r))},
j=1
has branching points at E1 , . . . , E2n+1 and ∞. Let a1 , . . . , an be the ovals in Γ lying above the open gaps ∆V1 , . . . , ∆Vn (i.e., aj = π −1 ∆Vj ). After Γ is cut along these ovals, it falls to two sheets Γ+ , Γ− , chosen in such a way that µ is positive on the upper edge of the cut [E2n+1 , E∞ ] in Γ+ . We supplement the ovals aj by n bcircles b1 , . . . , bn in such a way that the circles have the canonical intersection matrix: ai ◦ aj = bi ◦ bj = 0 and ai ◦ bj = δij . Next we take a basis dω1 , . . . , dωn of #holomorphic diﬀerentials on Γ, normalised by the conditions dωj , ak := ak dωj = 2πiδjk . These diﬀerentials exist and are uniquely deﬁned by the normalisation. The Riemann matrix B = B(r) = (Bjk ) of the curve Γ is deﬁned as the matrix of bperiods of the diﬀerentials dωj : Bjk = dωj , bk .
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Under our choice of the a, bcycles, the matrix B is real (see [BB, KK]). Its symmetric part is negatively deﬁned due to general properties of the Riemann matrices. Now we deﬁne the thetafunction θ of the curve Γ = Γ(r): 1 (B(r)s, s) + (z, s) , exp θ = θ(z; r) = 2 n s∈Z
where z ∈ C (the sum converges due to the properties of the Riemann matrix B). Clearly the function is 2πperiodic in imaginary directions: θ(z+2πiek ) = θ(z), where ek is the kth basis vector of Cn . The diﬀerentials dωj analytically depend on the parameter r ∈ Rn+ as well as the matrix B(r). Therefore the function θ(z; r) is analytic in r ∈ Rn+ . Since the matrix B is real, then θ is real and even: θ(z) = θ(z) and θ(z) = θ(−z). In particular, this function is real both in real and pure imaginary directions. Next, on the surface Γ(r) we consider Abelian diﬀerentials of the second kind dΩ1 , dΩ3 with vanishing aperiods and with the only poles at inﬁnity of the form √ dΩ1 = dk + (c + O(k−2 )) dk −1 , k = i λ → ∞, (4.2) dΩ3 = dk 3 + O(1) dk −1 , n
where c is an unknown constant. The normalisation (4.2) deﬁnes the diﬀerentials uniquely, see [S, NMP,BB]. Besides, dΩ1 =
i λn + . . . 3 λn+1 + . . . dλ, dΩ3 = − i dλ, 2 µ 2 µ
(4.2 )
where the dots stand for real polynomials of degree n − 1. Let us deﬁne complex nvectors iV(r) and iW(r) as the vectors of bperiods of these diﬀerentials: iVj = dΩ1 , bj ,
iWj = dΩ3 , bj .
The vector V is called the wavenumber vector and W – the frequency vector. The vector W is real and the vector V is integer. Moreover, V equals to the vector, formed by the numbers of the open gaps, see [BB, KK] (the latter vector was earlier denoted also as V ). One of top achievements of the ﬁnitegap theory is the Its–Matveev formula, which represents any ngap potential u(x) ∈ T n (r) in the form u(x) = u(x; r, z) = 2
∂2 ln θ(iVx + iz; r) + 2c. ∂x2
(4.3)
Here the constant c is the same as in (4.2). Since the meanvalue of the r.h.s. in (4.3) equals 2c, then we must have c = 0. The vector iz in (4.3)
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is iz = −A(D) − K , where K is the vector of Riemann constants (see [D, BB]) and A(D) is the Abel transformation of a positive divisor D = D(), D = D∞ . . . D\ , Dj ∈ aj . I.e., A(D) is a complex nvector such that its jth component A(D) equals A(D)j =
n r=1
Dr
∞
dωj ,
where {dωj } are the holomorphic diﬀerentials on Γ as above. The divisor D is a divisor of Dirichlet eigenvalues, i.e. Dj = (λj , µj ), where λj is an eigenvalue of the operator Lu subject to Dirichlet boundary conditions ϕ(0) = ϕ(2π) = 0 (each gap ∆j contains exactly one point from the Dirichlet spectrum, see [Ma, MT]).7 In particular, every point Dj analytically depends on u. The phase vector z turns out to be real, so θ(iVx + iz) is a real valued function of x. The thetafunction is nonzero at any imaginary point iξ ∈ iRn (see in [BB]). Since this function is periodic, then θ(iξ) ≥ C(r) > 0 for any real vector ξ. Hence, the r.h.s. of (4.3) is analytic in z ∈ T n . Due to the periodicity, we can treat z as a point in the torus T n . Thus we get an analytic map: T n (r) → T n , u(·) → z. This map has the analytic inverse given by the formula (4.3). The coordinate z on a ﬁnitegap torus T n (r) is called the thetaangles. Timeevolution u(t, x) of the ngap potential u(x) ∈ T n (r) as in (4.3) along the KdV ﬂow is given by the following formula, also due to Its–Matveev: u(t, x; r, z) = 2
∂2 ln θ(i(Vx + Wt + z); r) ∂x2
(4.4)
(we use that c = 0). ∂2 Denoting by G the function G(z, r) = 2 ∂x 2 ln θ(iVx + iz; r) x=0 , we write u as u(x; r, z) = G(z + V x, r). So the ngap torus T n (r) is represented in the form T n (r) = Φ0 (r, T n ) ⊂ H0d , where for any z ∈ T n , Φ0 (r, z) is the analytic function Φ0 (r, z)(x) = G(Vx + z, r). The formula (4.4) can be written as u(t, x; r, z) = Φ0 (r, z + W(r)t)(x). This shows that in the (r, z)variables the KdVﬂow on T 2n takes the form z˙ = W(r). I.e., the thetaangles z integrate the KdVequation on any torus T n (r). 7
This divisor can be also described as a divisor of poles of the Baker–Akhiezer eigenfunction ϕ(x; P ) of the operator Lu , Lu ϕ = π(P )ϕ, normalised at inﬁnity √ as ϕ ∼ ei λ x ; see [D, BB].
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Let R be a subcube of the octant Rn+ of the form R = {r ∈ Rn+  0 < rj < K} with some K > 0, and T 2n = Φ0 (R × T n ) ⊂ TV2n
(4.5)
for any wavenumber vector V. The set T 2n ⊂ H0d , d ≥ 1, is an invariant manifold of the KdV equation. It meets the assumptions i)–iii) from Section 2 since: The map Φ0 is an analytic embedding and T 2n is an analytic submanifold of H0d . The form Φ∗0 α2 is analytic and is nondegenerate for small r since it is nondegenerate at r = 0 by Theorem 3.1, so the set of its degeneracy is a proper analytic subset of the cube R (in fact, it is empty). The nondegeneracy assumption iv) also holds, as states the following Non Degeneracy Lemma: Lemma 6. The determinant det{∂Wj /∂rk } is nonzero almost everywhere. This result can be proven directly [BiK2, KK], or can be obtained as an immediate consequence of another lemma (proven in [BoK1, KK]): Lemma 7. For any ﬁnitegap manifold TV2n the corresponding frequency vector W as a function of r analytically extends to the origin and has there the 3 following asymptotic: Wj (r) = − 14 Vj3 + 8πV r2 + O(r4 ), j = 1, . . . , n. j j 4.3
Higher equations from the KdV hierarchy
Let us take any ngap manifold TV2n . The manifold itself and each torus T n (r) ⊂ TV2n are invariant for all Hamiltonian equations with the hamiltonians H0 , H1 , . . . from the KdVhierarchy (see Example 3.2). The ﬂow of any lth KdV equation on TV2n is very similar to the KdVﬂow: it is given by the thetaformula (4.4) where the frequencyvector W should be replaced by an nvector W(l) , formed by bperiods of some Abelian diﬀerential dΩ2l+1 . All results till Lemma 6, stated above for the KdV equation, have obvious reformulations for the higher KdV’s. An analogy for Lemma 7 is given by the following statement: The vector W(l) is analytic in r12 , . . . , rn2 and (l)
(l)
(l)
Wj (r) = Wj0 + Wj1 r2 + O(r4 )
(4.6)
(l)
for any j = 1, . . . , n, with some nonzero constants Wj1 . Any manifold TV2n treated as an invariant manifold of an lth KdV equation satisﬁes assumptions i)iv) for the same reason as for l = 1 (i.e., as in the KdVcase). 4.4
SineGordon equation under Dirichlet boundary conditions
Similar to the KdVcase, the equation (SG)+(EP) (see Example 3.3) has timequasiperiodic ﬁnitegap solutions with n basic freqencies which can be
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written in terms of thetafunctions. To symplify presentation we restrict ourselves to ﬁnitegap solutions with even number of gaps, so g = 2n everywhere below. These solutions can be written as ξ = (u, v)(t, x; D, r), where v = −u˙ and u(t, x) = 2i log
θ(i(V x + W t + D + ∆)) . θ(i(V x + W t + D))
(4.7)
Here θ is a thetafunction of the Riemann surface Γ, Γ = Γ(r) = {(λ, µ)  µ2 = λ
n
¯j )(λ − E −1 )(λ − E ¯ −1 )}, (λ − Ej )(λ − E j j
j=1
∆ is the vector (π, . . . , π), the wavenumber vector V and the frequency vector W are constructed in a way, similar to the KdVcase and D = (D1 , . . . , Dn , Dn , Dn−1 , . . . , D1 ),
D = (D1 , . . . , Dn ) ∈ Tn .
The vector r = (E1 , . . . , En ) is a point from some connected ndimensional real algebraic set R, which is a bounded part of the set Cn+ , C+ = {x + iy  y > 0}. The algebraic set R is “smooth near the real subspace” in the sence that for some δ0 > 0 the set R+ = {(E1 , . . . , En ) ∈ R  0 < Ej < δ0 ∀ j} is a smooth ndimensional real analytic manifold. Since the set of branching points of the surface Γ is inversioninvariant, then the vector W (the vector V ) turned out to be symmetric (antisymmetric) with respect to the involution T , T (U1 , . . . , U2n ) = (U2n , . . . , U1 ): TW = W ,
T V = −V .
˜ and V˜ , formed by the Hence, W and V are determined by the vectors W last n components: ˜ = (Wn+1 , . . . , W2n ) , W
V˜ = (Vn+1 , . . . , V2n ).
These vectors also will be called the frequency and wavenumber vectors. The wavenumber vector V˜ (r) is an rindependent integer vector (this guarantees that u is 2πperiodic in x). An important property of the family ¯ contains an unique real point r0 = of solutions we discuss is that the closure R 0 0 2n (E1 , . . . , En ) ∈ R and u → 0 when r → r0 (this convergence corresponds ¯j ] shrink to to a degeneration of the Riemann surface Γ when the gaps [Ej , E points). Moreover, u(0, x; D, r) =
n
! εj cos(Vn+j x + Dj ) + cos(−Vn+j x + Dj ) + o(ε2 ),
j=1
(4.8)
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where ε → 0 as r → r0 . For the sake of simplicity we restict ourselves to ﬁnitegap solutions such that ﬁrst n gaps are open in the sence that (Vn+1 , . . . , V2n ) = (1, . . . , n).
(4.9)
˜ is analytic on the algebraic set R and satisﬁes The frequency vector W the following analogies of Lemmas 6 and 7: ˜ (r)∗ is nondegenerate for a.a. r ∈ R; a) the tangent map W b) the set R+ admits analytic coordinates µ1 , . . . , µn such that Im Ej → 0 ˜ is analytic in µ and as µj → 0, the map W Wn+j (0) = 1 + j 2 =: j ∗ ,
(4.10) −16/j ∗ , j ≤ n, j = k, ∂Wn+j µ=0 = ∗ ∂µk −12/j , j = k. The set of timequasiperiodic solutions (4.7) is incomplete in the sence that the inﬁnitesimal solutions (4.8) do not contain the “cos 0x” term ε0 · 1. The reminding solutions are oddgap, they can be treated similar. For all these results see [BB], [BiK1, BoK2] and [KK]. A solution (4.7) is winding in a torus T n (r) = {ξ(0, ·; D, r)  D ∈ Tn }. " Altogether the tori form a set T 2n = r∈R T n (r) which is an image of Tn ×R under a map Φ0 , deﬁned in terms of the r.h.s. of (4.7) as in the KdVcase. The set T 2n and the map Φ0 satisfy the assumptions (i)–(iii) for the same trivial reasons as in the KdVcase, while assumption iv) follows from a).
5 5.1
Linearised equations and their Floquet solutions The linearised equation
Below (Z, α2 ) stands for a symplectic space (Z = Zd , α2 = J dz ∧ dz) with some ﬁxed d. We continue to study a quasilinear Hamiltonian equation u˙ = J∇H(u) = J(Au + ∇H(u)) =: VH (u),
(5.1)
where ord A = dA , ord ∇H = dH < dA , ord J = dJ and d ≥ dA /2. The equation is assumed to possess a 2ndimensional invariant manifold T 2n ⊂ Z as in Section 3, satisfying the assumptions i)iv). By iii), the regular part T02n of T 2n is ﬁlled by smooth solutions u0 (t) of (5.1) of the form u0 (t) = u0 (t; r0 , z0 ) = Φ0 (w0 (t)), where w0 (t) = (r0 , z0 + tω(r0 )) ∈ R0 × T n , t ∈ R. We linearise (5.1) about a solution u0 as above to get the nonautonomous linear equation v˙ = J(Av + ∇H(u0 (t))∗ v) =: JAt (t)v,
(5.2)
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which is our concern in this section. We recall that linear ﬂowmaps of equation (5.2) (if they exist) are denoted as Sτt ∗ (u0 (τ )) (see Deﬁnition 1), and supplement the assumptions i)–iv) assuming that (v) for any solution u0 of (5.1) in T02n the ﬂowmaps Sτt ∗ (u0 (τ )), −∞ < τ, t < ∞, are well deﬁned in the space Z = Zd . Since the equation (5.1) is Hamiltonian, then the ﬂowmaps Sτt ∗ (u) (u ∈ T02n ) are symplectomorphisms of the symplectic space (Z, α2 ) (see [KK]). To study equation (5.1) near T 2n we shall impose an integrability assumption on the linearised equation (5.2). Roughly speaking, this assumption means that the equation (5.2) has a complete system of timequasiperiodic Floquet solutions. Since the timeﬂow of (5.2) is formed by linear symplectomorphisms which preserve tangent spaces to T02n , then the ﬂow also deﬁnes symplectomorphisms of skeworthogonal complements Tu⊥0 T02n to the spaces Tu0 T02n in Tu0 Z ∼ Z.8 5.2
Floquet solutions
We call a solution v(t) of the equation (5.2) a Floquet solution if there exists a section Ψ of the complexiﬁed tangent bundle to Z, restricted to T02n (i.e., R0 × T n (r, z) → Ψ(r, z) ∈ TΦ0 (r,z) T02n ), and a complex function ν(r) such that the solution v has the form v(t) = v(t; r0 , z0 ) = eiν(r0 )t Ψ(w0 (t)),
w0 = (r0 , z0 + tω(r0 )).
(5.3)
It is assumed that v(t) solves (5.2) for any choice of r0 ∈ R0 and z0 ∈ Tn . We call ν(r) the (Floquet) exponent of a Floquet solution v. A Floquet solution v(t) is called a skeworthogonal Floquet solution if Ψ in (5.3) is a section of the complexiﬁed skewnormal bundle T ⊥c T 2n (its ﬁbres are complexiﬁcations of the spaces Tu⊥0 T02n ). Let us assume that equation (5.2) has an inﬁnite family of Floquet solutions v = vj (t). Clearly if vj is a Floquet solution, then v j is a solution with the exponent −ν j (r), corresponding to the section Ψj . We add this solution to the family; if a solution with the exponent −ν(r) already was there, we replace it by v j . Now the family is invariant with respect to the complex conjugation and the set of all exponents is invariant with respect to the involution ν → −ν. In addition we suppose that the set of exponents is invariant with respect to the complex conjugation ν → ν (this assumption holds trivially if all the frequencies are real); hence the set is invariant with respect to the involution ν → −ν. It is convenient to enumerate the Floquet solutions by integers from the set Zn = {±(n + 1), ±(n + 2), . . . }. We do it in such a way that ν−j (p) ≡ −νj (p) 8
Tu⊥0 T02n is formed by all vectors ξ ∈ Tu0 Z such that α2 (ξ, η) = 0 for each η ∈ Tu0 T02n .
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and Ψ−j ≡ Ψj if νj is real. So below we consider the following system of Floquet solutions : vj (t; r0 , z0 ) = eiνj (r0 )t Ψj (r0 , z0 + tω(r0 )), j ∈ Zn ; ν−j (r) ≡ −νj (r). (5.4) ˆ For each index k we denote by kˆ an index such that νkˆ = νk . Clearly kˆ = k for any k and kˆ = k if νk is real. We note that the hatmap is rindependent in any connected subdomain of R0 where all the functions νj (r) are diﬀerent. A frequency νj can have an algebraic singularity at the set Λj formed by all r such that νj (r) = νp (r) for some p = j. Situation becomes too intricate if there are inﬁnitely many nontrivial sets Λj . To avoid this complexiﬁcation we assume that a) there is a nonempty subdomain of R0 where νj = νk if j = k. Moreover, there exists j1 (depending on T 2n ) such that νj (r) = νk (r) for all r, all k and all j ≥ j1 , j = k. Since ν−j = −νj , then by this assumption νj = 0 if j ≥ j1 . The exponents νk with k ≥ j1 are assumed to be real analytic: b) for any k such that k ≥ j1 , νk is a realvalued analytic function on R (so νk ≡ −ν−k and Ψ−k = Ψk ). The section Ψk extends to an analytic map Πc × { Im z < δ} → Z c and νk extends to an analytic function on Πc . In particular, kˆ = k if k ≥ j1 . For sophisticated integrable equation like the SG equation some exponents νk (r) with k < j1 have nontrivial algebraic singularities. The assumptions we shall impose now on the exponents νk with k < j1 (below we call such k small ), are made ad hoc – they are met by Floquet solutions of Laxintegrable equations. These assumptions are empty for integrable equations with seladjoint Loperators, e.g., for the KdV equation. Let us denote M = j1 − n − 1. c) The functions νj with small j are continuous in R and have the form νj (r) = ν˜(λj (r), r), where {λj (r)} are branches of some 2M valued algebraic function, deﬁned on Πc , and ν˜ is an analytic complex function. By c) the functions νj with small j are algebraic (as well as the functions λj ). They are analytic in Πc outside a discriminant set D of the algebraical function ν˜. We note that D ∩ R is a proper analytic subset of R since no two exponents νj , νk coincide identically in R by the assumption i). The set D ∩ Rc contains algebraic singularities of the Floquet exponents and is contained in the set Λ = {r ∈ Rc  νj (r) = νk (r) for some j = k, with j, k < j1 }, which is a proper analytic subset of Rc . It contains zeroes of the exponents νj since they are odd in j. We add Λ to the singular set Rsc : Rsc := Rsc ∪ (Λ ∩ Rc ),
Rs := Rs ∪ (Λ ∩ R) ,
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and modify the regular set R0 accordingly. Example. Eigenvalues {λj } of a real matrix B(a) which analytically depends on a real vectorparameter a are zeroes of the characteristic equation det(B(a) − λE) = 0 and are algebraic functions of a. A priori they have singularities at the sets Λjk = {λj = λk }. Some of these singularities can be removed by reenumerating the eigenvalues before or behind the sets Λjk . In particular, if the matrix B(a) is symmetric, then under proper enumeration the eigenvalues have no singularities at all (this is Rellich’s theorem, see [Kat2,RS4]). However, if λj and λk are real “before” Λjk and have nontrivial imaginary parts “behind” Λjk , then a singularity at this set is unremovable. Now we pass to smoothness of the sections Ψj with small j (j < j1 ): $ z, λ) ∈ Z c , such that d) There exists a bounded analytic map (r, z, λ) → Ψ(r, n $ Ψj (r, z) = Ψ(r, z; λj (r)) for (r, z) ∈ R0 × T and all small j. This assumption agrees with smoothness of eigenvectors in ﬁnitedimensional spectral problems: j Example (continuation). Let us denote by B j = (Blm  1 ≤ l, m ≤ n) the matrix B j (a) = B − λj (a)E, so " Bξ = λj ξ if B j ξ = 0. Let us assume that rk B j (a) = n − 1 for a ∈ / Λj = k Λjk and denote by ξm (a) the algebraic j j (a) in complement to the element Bnm the jmatrix B . Then the vector ξ = (ξ1 , . . . , ξn ) is nonzero for a ∈ / Λj and m Blm ξm = 0 since: for l = n the sum equals det B j = 0 and for l = n it vanishes by an elementary linear algebra. The vector ξ is an eigenvector of B. It is a polynomial in the eigenvalues λj and in the elements of the matrix B. It vanishes at Λj .
5.3
Complete systems of Floquet solutions
Let us take any basis {ϕj  j ∈ Z0 } of the Hilbert scale {Zs } and assume that the basis is symplectic, i.e., α2 [ϕj , ϕ−k ] = Jϕj , ϕ−k =
δj,k νjJ
for all j, k ∈ Z0 ,
(5.5)
J and νjJ > 0 if j is positive. Due to (5.5), Jϕk = ϕ−k /νkJ where νjJ = −ν−j for any k ∈ Z0 . Since J is an anti selfadjoint isomorphism of the scale {Zs } of order −dJ ≤ 0, then C1−1 j dJ ≤ νjJ ≤ C1 j dJ for every j ≥ 1 with some C1 ≥ 1. For any real s we denote by Ys the Hilbert space
Ys = span{ϕj  j ∈ Zn } ⊂ Zs . The spaces {Ys , α2 Ys } form a symplectic Hilbert scale with the basis {ϕj  j ∈ Zn }. In Y0c we choose some complex basis {ψj  j ∈ Zn } and assume that it is symplectic: α2 [ψj , ψ−k ] = δj,k µj .
(5.6)
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Besides, we assume that for big j it agrees with the basis {ϕj }: √ µj = i/νjJ and ψ±j = (ϕj ∓ iϕ−j )/ 2 if j ≥ j1 .
(5.7)
Example. If {ϕj } is the trigonometric basis as in (1.1), i.e. ϕk = π −1/2 cos kx and ϕ−k = −π −1/2 sin kx, then for k ≥ j1 the functions ψk are complex exponents, ψk = (2π)−1/2 eikx . Let {vj } be a system of Floquet solutions as in Section 5.2 and {Ψj } are corresponding sections. For any (r, z) ∈ R0 ×T n we denote by Φ1 (r, z) a linear map from Y c = Ydc to Z c which identiﬁes ψj with Ψj : Φ1 (r, z) : Y c → Z c ,
ψj → Ψj (r, z) ∀j ∈ Zn .
(5.8)
The map Φ1 will be used to formulate an important assumption of completeness of a system of Floquet solutions. Before to do this we cut out of the set R0 some “neighbourhood of inﬁnity” and any neighbourhood of the singular set Rs to get an open domain R1 , R1 R0 \ Rs . Possibly, R1 is disconnected. To simplify notations we assume that R1 belongs to a single chart of the analytic manifold R0 and treat R1 as a bounded domain in Rn . We ﬁx any bounded complex domain R1c which contains R1 with its δneighbourhood and does not intersect the singular set Rsc . We denote by W1 the set W1 = R1 × T n and denote by W1c its complex neighbourhood, W1c = R1c × {Im z < δ}. Deﬁnition 3. A system of Floquet solutions (5.4) which satisﬁes the analyticity assumptions a)–d) is called complete (in the space Z = Zd ) if: 0) it is formed by skeworthogonal Floquet solutions, and for any (r, z) ∈ R0 × T n we have: 1a) the functions βj = iα2 [Ψj (r, z), Ψ−j (r, z)], j ∈ Zn , are zindependent: βj = βj (r), b) there is a nonempty subdomain of R where no function βj (r) vanishes identically, c) the vectors {Ψj (r, z)} form a skeworthogonal system in the space TΦ⊥c T 2n , that is: 0 (r,z) α2 [Ψj , Ψ−k ] = iβj (r)δj,k
∀j, k .
(5.9)
2) The vectors {Ψj (r, z)} are uniformly asymptotically close to the complex basis {ψj } and the exponents νj (r) are close to constants. Namely, a) the linear map Φ1 (r, z) equals to the natural embedding ι Y → Z up to a ∆smoothing operator, ∆ > 0: Φ1 (r, z) − ιd,d+∆ ≤ C1
for all (r, z) ∈ W1c ;
(5.10)
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b) for large j the functions βj (r) in (5.9) are analytic in R1c and are there close to the constants (νjJ )−1 , deﬁned in (5.5) (cf. (5.7)): βj (r) − 1/νjJ  ≤ C2 j−κ
for r ∈ R1c
(5.11)
with some κ ≥ max(1, dJ + ∆); c) the exponents νj are bounded in R1c and are there “asymptotically close to constants”. That is, νj (r) ≤ C3 jdA +dJ for r ∈ R1c and
∇νj (r) ≤ C4 j∆
for r ∈ R1c
(5.12)
$ < d A + dJ . with some real ∆ The constants C1 − C4 in this deﬁnition may depend on the domain R1 but not on j. Since Ψ−j (w) = Ψj (w) for real w and big j, then the corresponding functions βj are real and β−j ≡ −βj . As µj ≥ C −1 j −dJ , then by the assumption (5.11) we have that βj (r) ≥ 12 µj for all r ∈ R1c and j > j2 , with some new constant j2 . We consider the product $b(r) =
j2
βj2 (r).
j=n+1
It follows from the properties c) and d) that the function ˜b is analytic in Πc (see [KK]). Due to 1b) a set of its zeroes in Rc is a proper analytic subset. We add it to the complex singular set Rsc and accordingly modify the sets Rs and R0 . If it is necessary, we also decrease the domain R1 so that the inclusion R1 R0 \ Rs still holds true. Remark. The set Rs as it is deﬁned now is the ﬁnal singular set for our constructions. It comprises: 1) the singular part of the algebraic set R, 2) the set of degeneracy of the pullback symplectic structure Φ∗0 α2 , 3) algebraic singularities of the Floquet exponents and 4) points where any two of them coincide. Finally, it contains 5) the zeroset of the function ˜b we have just constructed. The last set is a set of degeneracy of the system {Ψj } since a vector Ψj (r, z) is skeworthogonal to the tangent space Tu T 2n and to all the vectors Ψk (r, z) as soon as βj (r) = 0. In the same time in Lemma 8 below we prove that the vectors {Ψj } form a basis of the skeworthogonal space Tu⊥c T 2n if r ∈ / Rs . ˜ \ Rs may be chosen to occupy most part of R0 in the The set R1 R ˜ is a bounded chart of the manifold R0 , mesn is the sense of measure: If R ˜ and γ is any positive number, then R1 can be chosen Lebesgue measure in R in such a way that ˜ \ R) ≤ γ. mesn (R
(5.13)
A complete system of skeworthogonal Floquet solutions spans the skeworthogonal spaces Tu⊥c T 2n in conformity with the term “complete” we use:
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Lemma 8. For any w ∈ W1c the map Φ1 (w) deﬁnes an isomorphism of the spaces Y c and Tu⊥c T 2n , where u = Φ0 (w). 9 Proof. By (5.10) the map Φ1 (w) is a compact perturbation of the embedding ι Y c → Z c , so ind C Φ1 (w) = ind ι = 2n. As a range of Φ1 lies in Tu⊥c T 2n , then dimC Coker Φ1 ≥ 2n. So if we can show that Ker Φ1 = {0}, then the range of Φ1 equals Tu⊥c T 2n and the assertion will follow. Suppose that the ψ kernel is nontrivial. Then it contains a nonzero vector ξ = y j j and we have 0 = Φ1 ξ = yj Ψj (w). The skewinner product of the righthand side with any vector Ψ−j (w) equals i yj βj (r) (see (5.9)). Thus, yj ≡ 0 and ξ = 0. Contradiction. Decreasing in a need the complex neighbourhood W1c of W1 we easily get the following result: Lemma 9. For any s ∈ [−d − dJ − ∆, d + ∆] the operator Φ1 (w) : Ysc → Zsc analytically depends on w ∈ W1c and is uniformly bounded. Moreover, for any c is analytic in w ∈ W1c as well. s as above the map Φ1 (w) − ι Ysc → Zs+∆ Example ((Birkhoﬀintegrable systems, see [K1, p. 401). and [Kap, BKM])] Let Z be a space of sequences ξ = (x1 , y1 ; x2 , y2 ; . . . ), given some Hilbert norm and given the “usual” symplectic structure by means of the 2form J dξ ∧ dξ, where J(x1 , y1 ; . . . ) = (−y1 , x1 ; . . . ). We denote pj = (x2j + yj2 )/2, qj = Arg(xj + iyj ) and consider an analytic hamiltonian h(p1 , p2 , . . . ). The subspace T 2n ⊂ Z formed by all vectors ξ such that 0 = xn+1 = yn+1 = . . . is invariant for the Hamiltonian vector ﬁeld Vh and the restricted to T 2n system is obviously integrable. Let us denote pn = (p1 , . . . , pn ), q n (t) = (q1 , . . . , qn ) n ) and denote by νj the functions νj (pn ) = ∂h(p∂p,0,... , j ≥ 1. We shall identify j n n p with the vector (p , 0, . . . ). The manifold T 2n is ﬁlled with solutions ξ(t) = {pn = const , q n = tν n (pn ) + ϕn ; pr = 0 for r > n}, where ϕn ∈ T n and ν n = (ν1 , . . . , νn ). For any j > n let us consider a smooth variation ξ(t, ε) of a solution ξ(t), which changes no action pl except pj and makes the latter equal ε2 : ξ(t, ε) = {pn (t) = pn , q n (t) = tν n (pn )+q0n (ε); xl (t) = yl (t) = 0 if l > n, l = j} and xj (t) = ε cos(tνj (pn )+ϕ(ε)), yj = ε sin(tνj (pn )+ϕ(ε)) , where ϕ(ε) ∈ S 1 . ∂ ξ(t, ε) ε=0 is a solution of the equation, linearised about The curve v˜j = ∂ε ξ(t). It equals v˜j (t, ϕ) = {δpn = 0, δq n = q0n (0); δx(t), δy(t)}, 9
For a complex w and u = Φ0 (w) we deﬁne Tu⊥c T 2n as the set of all z ∈ Z c such that α2 [z, Φ0 (w)∗ ξ] = 0 for any ξ ∈ Tw W1c .
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where δxr (t) = δyr (t) = 0 if r = j and δxj = cos(tνj (pn ) + ϕ), δyj = sin(tνj (pn ) + ϕ),
ϕ = ϕ(0).
The curve vtriv (t) = {δpn = 0, δq n = q0n (0); δx = δy = 0} is a trivial solution of the linearised equation (it may be obtained using a variation of ξ(t), corresponding to a rotation of the phasevector q n ). An appropriate complex linear combination of the solutions v˜j (t, 0), v˜j (t, π) and of the trivial n solution as above takes the Floquet form vj (t) = eiνj (p )t Ψj , where Ψj = (0, . . . ; i, 1; 0, . . . ) (the pair (i, 1) stands on the jth place). Let us suppose that νj  ≤ Cj dA for some dA and that (5.12) holds. Then the system of Floquet solutions {vj , vj  j ≥ n + 1} is complete in the sense of Deﬁnition 3. This example illustrates well the deﬁnition but it is too simple and so too restrictive: to be Birkhoﬀ integrable a ﬁnitedimensional system has to have dim Z/2 integrals of motion, but to have a complete system of Floquet solutions for the equations linearised about solutions in T 2n it needs only n of them (see below Proposition 2). To be useful in analytical studies of the equation (5.1) and its perturbations, a system of Floquet solutions should be complete and nonresonant: Deﬁnition 4. A system of Floquet exponents {νj (r)  j ∈ Zn } satisfying a)–c) is called nonresonant if: 3) there exists a domain O ⊂ R0 such that for all s ∈ Zn and all j, k ∈ Zn , j = −k, we have: ω(r) · s + νj (r) ≡ 0 in O, ω(r) · s + νj (r) + νk (r) ≡ 0 in O.
(5.14) (5.15)
The system of Floquet solutions with nonresonant exponents also is called nonresonant. Since the exponents νj are algebraic (or analytic) functions, then zeroset of any resonance is nowhere dense. Finally we give: Deﬁnition 5. A system of Floquet solutions (5.4) satisfying a)–d) is called complete nonresonant if it satisﬁes assumptions 0)–3) from Deﬁnitions 3, 4. It turns out that the assumption 1) follows from 3): Lemma 10. Any nonresonant system of Floquet solutions satisfy assumptions 0),1a) and 1c) from Deﬁnition 3. Proof. To check 1c) we should prove that for any j = −k the function F (r, z) = α2 [Ψj , Ψk ] vanishes identically. To do it let us consider the function f (t; r, z), f := ei(νj +νk )t α2 [Ψj (w(t)), Ψk (w(t))] = α2 [vj (t), vk (t)],
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where w(t) = (r, z + tω(r)). Since the ﬂowmaps Stt12∗ are symplectic [KK], then the skewproduct of any Floquet solutions vj and vk is timeindependent and df %% = i(νj + νk )F + z F · ω. 0= % dt t=0 Let us expand F as Fourier series, F = eis·z F&(r, s). From the last identity we get that F&(r, s)(νj + νk + s · ω(r)) = 0 for all s and all r. Since the second factor is nonzero for almost all r, so F&(r, s) ≡ 0 and F (r, z) ≡ 0. If j = −k, then we have: const ≡ α2 [vj (t), v−j (t)] = α2 [Ψj (w(t)), Ψ−j (w(t))]. Because the assumption iv) (Section 3), the curve w(t) is dense in a torus {r} × Tn for almost all r. So the lefthand side of (5.9) with k = −j is zindependent for almost all r. By continuity, it is zindependent for all r, as states 1a). To check 0) one has to argue similar, using variations δr, δz of the initial conditions for the curve w(t). Corollary. A system of Floquet solutions (5.4) which meets the assumptions a)–d) from Section 5 as well as the assumptions 2), 3) from Deﬁnitions 3, 4 is skeworthogonal to T 2n and is complete nonresonant, provided the assumption 1b) holds. The latter happens e.g., if there exists a point r∗ ∈ R such that Ψj (r, z) → ψj as r tends to r∗ . Here R signiﬁes the closure of R in RN where R is a subset. Practically the point r∗ corresponds to the zerosolution of the equation (5.1) (or another trivial solution). This result simpliﬁes veriﬁcation of completeness for a system of Floquet solutions since it is much easier to check the nonresonance relations (5.14), (5.15) than the completeness 1a)–1c). The transformation Φ1 integrates the linearised equation (5.2): it sends the curves yj = eiνj (r0 )t ψj to solutions vj (t) of (5.2). It is convenient to have this transformations symplectic. For this end the sections {Ψj } have to be properly reordered and normalised by multiplying by some analytic functions of r; simultaneously the basis {ψj } also have to be transformed by a linear symplectomorphism which changes ﬁnitely many its components only. In this way the following result can be proven: Proposition 2. Given any complete system of Floquet solutions (5.4) we can normalise the sections {Ψj } and the basis {ψj } is such a way that the new basis still meets (5.6), (5.7) and the new system of Floquet solutions still is complete. Besides, a) for any (r, z) ∈ W1 = R1 × Tn the map Φ1 (r, z) deﬁnes a symplectic ⊥ T 2n which analytically in (r, z) ∈ W1c extends isomorphism of Y and TΦ(r,z) c to a bounded linear map Y → Z c ; b) the nonautonomous linear map Φ1 (r, z+tω(r)) sends solutions of an autonomous equation y˙ = JB(r)y, y ∈ Y c , to solutions of (5.2). The selfadjoint
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operator B(r) is analytic in r. Besides, ord B(r) ≤ dA and ord ∇r B(r) ≤ $ Spectrum of the operator JB(r) equals to the set of Floquet expo−dJ − ∆. nents of the solutions (5.4). The basis {ψj } may depend on a connected component of the set R1 . The leading Lyapunov exponent of equation (5.2) in Zd is a number a equal to the supremum over all real numbers a such that lim t→∞ e−a t v(t)d = ∞ for some solution v(t) ⊂ Zd of (5.2). A solution u0 (t) of (5.1) is called linearly stable if the leading Lyapunov exponent of the corresponding linearised equation (5.2) vanishes. A direct consequence of Proposition 2 is the following Corollary. If the linearised equations (5.2) have complete system of Floquet solutions, then the leading Lyapunov exponent of the equation corresponding to a solution u0 = u0 (t; r, z) with r ∈ R1 equals ν I (r) = max{Im νj (r)  n < j < j1 }.10 Indeed, by the proposition any variation u (t) of a solution u0 (t) can be written as Φ0 (u0 )∗ (r , z ) + Φ1 (u0 )y and in terms of the primevariables the equation (5.2) reads as r˙ = 0,
z˙ = ω(r)∗ r ,
y˙ = JB(r)y .
(5.16)
Decomposing y (0) in the basis {ψj } we ﬁnd that e−at u (t)s → 0 as t grows, if a > ν I (r). If a < ν I (r) and ψj is an eigenvector of JB(r) with the eigenvalue νj such that Im νj = a, then y (t) = e−iνj t ψ−j is the y component of a solution of (5.16). A norm of this solution grow with t faster than eat . 5.4
Lowerdimensional invariant tori of ﬁnitedimensional systems and Floquet’s theorem
Let O be a domain in the Euclidean space R2N , given the usual symplectic structure. Let H1 , . . . , Hn , 1 ≤ n < N , be a system of commuting hamiltonians, deﬁned and analytic in O. Let T n ⊂ O be a torus, analytically embedded in O, which is invariant for all n Hamiltonian vector ﬁelds VHj . The vector ﬁelds are assumed to be linearly independent at any point of the torus. Under mild nondegeneracy assumptions on the system of hamiltonians (see [Nek]), the torus T n can be proven to belong to an ndimensional family of invariant ntori Trn : Trn , 0 ∈ R Rn ; T n = T0n , T n ⊂ T 2n = r∈R
where T 2n is an analytic 2ndimensional submanifold of O. Moreover, the symplectic form, restricted to T 2n , is nondegenerate and T 2n admits analytic 10
We recall that the functions νj (r) with j ≥ j1 are real valued by the assumption b).
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coordinates (r, z), z ∈ Tn , such that for every j = 1, . . . , n the vector ﬁeld VHj , restricted to T 2n , takes the form l ωjl (r)∂/∂zl (the functions ωjl (r) all are analytic). Instead of presenting here the nondegeneracy assumptions, we just assume existence of a family of invariant ntori as above. Then for any r there exist linear combinations K1 , . . . , Kn of the original hamiltonians Hj such that for every j the vector ﬁeld VKj restricted to the torus Trn equals ∂/∂zj . Accordingly, at any point (r, z) ∈ Trn every vector ﬁeld VKj deﬁnes N − n j Floquet multipliers eiλl (r) , l = 1, . . . , N − n, corresponding to directions, transversal to T 2n . 11 For simplicity we assume that T 2n is a linearly stable invariant set of every vector ﬁeld VKj (so also of every VHj ). Then all the functions λjl (r) are real. The following result is a version of the Floquet theorem “for multidimensional time”. For a proof see [K3] and [KK]. Proposition 3. Under the given above assumptions, every vector ﬁeld VHj , linearised about its solutions in T 2n , has a complete system of N − n skeworthogonal Floquet solutions with real exponents νj (r). We note that in the ﬁnitedimensional situation which we discuss now, the item 2) of Deﬁnition 3 becomes trivial.
6
Linearised Laxintegrable equations
Now we pass to the problem of constructing complete systems of Floquet solutions of inﬁnitedimensional sustems. If (3.1) was a ﬁnitedimensional system (i.e., dim Z < ∞) with an integrable subsystem T 2n as above, then by Proposition 2 linearised equation (5.2) would have a complete system of Floquet solutions provided that the equation (3.1) had n nondegenerate integrals in involution. For inﬁnitedimensional systems the Floquet theorem is unknown. In this section we show that for Laxintegrable equations Floquet solutions can be constructed as quadratic forms of the eigenfunctions of the corresponding Loperator. 6.1
Abstract situation
Let u(t) be any smooth solution of a Laxintegrable equation (5.1)=(2.6). % % ∂ For any smooth vector v we denote Lt (v) = Lu(t) (v) = ∂ε Lu(t)+εv % , and ε=0
11
The multipliers are deﬁned as eigenvalues of the linearized time2π ﬂowmap of the vector ﬁeld VKj , restricted to a skeworthogonal component to the space T(r,z) T 2n . They are zindependent, see [K3].
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similar deﬁne operators At (v). Diﬀerentiating equation (3.4) in a direction v, we get a Laxrepresentation for the linearised equation (5.2): d L (v) = [At (v), Lt ] + [At , Lt (v)], dt t where At = Au(t) and Lt = Lu(t) . Let us consider smooth eigenvectors of the operator L0 = Lu0 and of its conjugate operator L∗0 , corresponding to the same eigenvalue λ: L0 χ0 = λχ0 , L∗0 ξ0 = λξ0 . We assume that the following initialvalue problems, χ(t) ˙ = At χ(t),
χ(0) = χ0 ,
˙ = −A∗ ξ(t), ξ(t) t
ξ(0) = ξ0 ,
(6.1)
have smooth solutions χ(t) and ξ(t). Then for any t we have Lt χ(t) = λχ(t) and L∗t ξ(t) = λξ(t) (see Lemma 2.3 for the proof of the ﬁrst relation; proof of the second is identical). We claim that d L (v(t))χ, ξ = 0. (6.2) dt t Indeed, abbreviating Lt (v(t)) to L and At (v(t)) to A , we write the lefthand side of (6.2) as ˙ + L˙ χ, ξ + L χ, L χ, ξ ˙ ξ = L χ, −A∗ ξ + ([A , L] + [A, L ])χ, ξ + L Aχ, ξ = [A , L]χ, ξ = A Lχ, ξ − A χ, L∗ ξ = (λ − λ)A χ, ξ = 0. Since Lt (w) linearly depends on w ∈ Zs as an operator from Zs to Zs −d , then Lt (w)χ, ξZ = w, qt (χ, ξ)Z
∀ w,
(6.3)
where qt (χ, ξ) = qu(t) (χ, ξ) is an Z−s valued quadratic form of χ, ξ ∈ Zs , which is C 1 smooth in t. Hence, we can rewrite (6.2) as d v(t), qt (χ, ξ) ≡ 0. dt For a moment let us denote qt (χ, ξ) = w. Then v, At Jw = −JAt v, w = −v, ˙ w = v, w, ˙
(6.4)
(6.5)
where the last equality follows from (6.4). At this point we assume that the ﬂowmaps Sτt ∗ (u(τ )) of the linearised equation (5.2) preserves the space Z∞ . Then the set {v(t)} formed by values at time t of all smooth solutions of equation (5.2) equals Z∞ , so w˙ = A(t)Jw since (6.5) holds for any t and v(·). Therefore J w˙ = JA(t)Jw, i.e. the curve Jw(t) = for all solutions J q(χ(t), ξ(t)) satisﬁes the equation (5.2). Thus, linearised Laxintegrable equations have solutions which can be obtained as bilinear forms of eigenfunctions of the Loperator and its adjoint:
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Theorem 3. If ﬂowmaps of the linearised equation (5.2) preserve the space Z∞ and the curves χ(t), ξ(t) are smooth solutions of equations (6.1), then the function J qt (χ(t), ξ(t)) with qt deﬁned in (6.3) solves the linearised equation (5.2). Remark. Let {Zs } be a Sobolev scale of 2πperiodic functions of a spacevariable x. If the ﬂows of linear equations (5.2) and (6.1) preserve Sobolev spaces of 2πantiperiodic functions, the curves χ(t), ξ(t) are constructed as above and (6.3) holds with some 2πperiodic function qt (χ, ξ)(x), then due to the same arguments as before the function J(q) solves (5.2). Remarkably the solutions given by the theorem and the remark have the Floquet form (5.3) and jointly form a complete nondegenerate family. Below we check this property for the KdV and SG equations. 6.2
Linearised KdV equation
Now we consider the KdV equation and take for the invariant manifold T 2n a bounded part of any ﬁnitegap manifold TV2n of the form (4.5). We have already checked that it satisﬁes assumptions i)–iv) from Section 5. For any ngap solution u0 (t, · ) ∈ TVn (r) the equation linearised about u0 takes the form v˙ =
1 3 ∂ vxxx + (u0 (t, x)v). 4 2 ∂x
(6.6)
Since u0 (t, x) is a smooth function, then this equation is welldeﬁned in Sobolev spaces H0d with d ≥ 1. Thus the assumption v) on the invariant manifold also is satisﬁed. The equation (6.6) has trivial solutions, obtained by diﬀerentiations in directions, tangent to the ngap tori. They can be written as ∂u(t,x;r,z) (see ∂zj (4.4)). We recall that the Loperator of the KdV equation is the Sturm–Liouville operator L = −∂ 2/∂x2 − u0 (t, x) and consider any its complex eigenfunction χ(x; λ) with an eigenvalue λ, satisfying the Floquet–Bloch boundary conditions: Lχ(x; λ) = λχ(x; λ),
χ(x + 2π; λ) = eiρ χ(x; λ), ρ = ρ(λ).
(6.7)
One of the most important and elegant properties of the KdV equation (and of the whole class of Laxintegrable equations) is that χ as a function of λ is meromorphic in Γ \ ∞ (Γ = Γ(r) is the Riemann surface deﬁned in√Section 4) and can be normalised to have at inﬁnity the singularity exp i λ x. This function admits a representation in terms of the same thetafunction θ and the vectors V , W , z as in Section 4. The representation is given by the
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ItsMatveev formula (see in [DMN,D,BB]): χ(t, x; r, z; P ) = eΩ1 (P )x+Ω3 (P )t
θ(A(P ) + i(V x + W t + z))θ(iz) , θ(A(P ) + iz)θ(i(V x + W t + z)) P = (λ, µ) ∈ Γ.
Here A(P ) is the Abel transformation, the same as above, and Ω1 , Ω3 are Abel integrals of the diﬀerentials dΩ1 , dΩ3 . The integrals are deﬁned modulo periods of the diﬀerentials. For P = (λ, µ) with real λ we normalise the integrals in the following way:
λ
Ωj (λ, µ) =
dΩj
for (λ, µ) ∈ Γ+ , λ ∈ R,
E1
where [E1 , λ] stands for the path in Γ+ through upper edges of the cuts. We denote by σ the holomorphic involution of Γ which transposes the sheets, σ(λ, µ) = (λ, −µ). Denoting for any P = (λ, µ) ∈ Γ by γP the path from σ(P ) to P through E1 , equal to γP = σ(−[E1 , λ]) ∪ [E1 , λ], we get that Ωj (P ) =
1 2
dΩj , γP
since σ ∗ dΩj = −dΩj due to the normalisation (4.2). Now we take a point P close to inﬁnity and denote by µP the path from σ(P ) to P equal to a lift to Γ of the circle in Cλ centred at inﬁnity, which passes through λ and is cut there. The loop γP − µP is contractible in Γ since it envelops all the cuts, so Ωj (P ) = 12 µP dΩj . Using this equality and (4.2) (with c = 0) we get the asymptotics: √ Ω1 (P ) = k + O(k −2 ), Ω3 (P ) = k 3 + O(k −1 ), k = i λ. (6.8) Let us denote f (U ; r, z; P ) =
θ(A(P ) + iU + iz)θ(iz) θ(A(P ) + iz)θ(iU + iz)
(6.9)
and rewrite χ as χ(t, x; r, z; P ) = eΩ1 (P )x+Ω3 (P )t f (V x + W t; r, z; P ).
(6.10)
By the Riemann theorem (see [D,BB]) the ﬁrst term of the denominator in the righthand side of (6.9) has exactly n zeroes which form poles of the function P → f and lie in the ovals a1 , . . . , an . Since θ(iU + iz) ≥ C(r) > 0 and θ(A(∞) + iz) ≥ C(r), then the function f (U ; r, z; P ) is analytic and bounded for r from an appropriate complex neighbourhood of any compact
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subset of R and for (λ, z, U ) from the complex domain { Im λ, Im z, Im U  < δ, Re λ > E1 and dist (λ, ∆Vj ) > δ ∀ j} ,
(6.11)
where δ > 0 is suﬃciently small. We recall that the closed gaps [λ2j−1 = λ 2j ] are labelled by indices j ∈ NV . So for any P = P±j , where P±j = {± R(λ2j ), λ2j } ∈ Γ, the function χ(t, x; P±j ) must be 4πperiodic in x. Since f is 2πperiodic, we should have Ω1 (P±j ) ∈ 2i Z. This relation holds identically in r. When r tends to zero, λ2j tends to j 2/4 and Ω1 (Pj ) tends to ij/2 (this follows from elementary calculations, see [KK]). Therefore, Ω1 (Pj ) =
i j, 2
j ∈ NV .
(6.12)
Conversely, for any P which meets (6.12) the function (6.10) is 4πperiodic. Since the operator A for the KdV equation is anti selfadjoint, then the second equation in (6.1) coincides with the ﬁrst and we can take ξ(t) = χ(t). Now the quadratic form q as in Theorem 3 equals χ2 . Finally, since J = ∂/∂x, then the solutions of the linearised equation (5.2)=(6.6) constructed in Theorem 3 are curves vj (t) ∈ Z of the form vj (t, x; r, z) =
(2π)−1/2 2Ω1 (Pj )
∂ e2(Ω1 (Pj )x+Ω3 (Pj )t) f 2 (V x + W t; r, z; Pj ) . ∂x (6.13)
Here j ∈ ZV , Pj = Pj (r) and the ﬁrst factor in the righthand side is a convenient normalisation. Thus we have obtained a system of Floquet solutions of the form (5.4),12 where the sections Ψj of the bundle T c H0d T 2n have the form 2Ω1 (Pj )x e ∂ √ (6.14) Ψj (r, z)(x) = f 2 (V x; r, z; Pj ) , j ∈ ZV , ∂x 2 2π Ω1 (Pj ) P and the exponents νj are νj (r) = −2iΩ3 (Pj ) = −2i E1j dΩ3 . Since the diﬀerential i dΩ3 is real (see (4.2 )), then the exponents νj (r) are real for real r and are analytic in r (they have no algebraic singularities). We claim that this system is complete nonresonant. To simplify notation we suppose that V = (1, . . . , n). √ Now the complex basis {ψj  j ∈ Z0 } is the exponential basis ψj = eijx/ 2π. The system of Floquet exponents is nonresonant. To prove the nonresonance we may assume that the vector r is suﬃciently small. For any 12
the set of indices ZV which we use now is in obvious 11 correspondence with the set Zn .
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j ∈ ZV = Zn we denote by V (n+1) the (n + 1)vector (V , j) and view the torus TVn (r) as a degenerate (n + 1)gap torus TVn+1 (n+1) (r, 0). It can be checked (n+1)
[KK] that the integral which deﬁnes νj (r) equals Wn+1 (r, 0). Since the frequency vector ω for ﬁnitegap solutions which ﬁll the torus TVn (r) is ω = W , then the nonresonance relation (5.18) which has to be checked takes the form n
(n+1)
Wl
(n+1)
(r, 0)sl + Wn+1 (r, 0) ≡ 0.
(6.15)
l=1
We can suppose that s = 0; say, s1 = 0. By Lemma 7, for r = (ε, 0, . . . , 0) (n+1) we have: Wl = const + δl,1 8V3 1 ε2 + O(ε4 ). Therefore, the lefthand side of (6.15) equals const + s1 8V3 1 ε2 + O(ε4 ). It does not vanish identically and (6.15) follows. The system is complete. By the Corollary to Lemma 10 we should only check the assumptions 1b) and 2) from Deﬁnition 3. The function f ( · ; r, z; P ) converges to unit as r → 0 (as well as the thetafunction, see [KK]). Therefore Ψj (x) converges to the complex exponent (2π)−1/2 eijx = ψj (x), so 1b) follows and it remains to check the item 2). Given any γ > 0 we ﬁx a subset R1 R such that mes(R \ R1 ) < γ (see (5.13)). For r ∈ R1 we shall check the properties 2a)–2c). First we show that the map Φ1 deﬁned in (5.8) is close to the embedding ι. Since Ψ−j = Ψj , we have to examine the vectors Ψj with j ∈ NV only. Since the potentials u(x) have zero meanvalues, then λ(Pj ) = 14 j 2 + O(j −1 ) √ (see [Ma,MT]) and k(Pj ) = i λ = 2i j + O(j −2 ). Using (6.8) we get that Ω3 (Pj ) = − 8i j 3 + O(j −1 ). So νj (r) = − 14 j 3 + O(j −1 ),
(6.16)
uniformly in r from a complex neighbourhood R1 + δ of R1 . Since the holomorphic diﬀerentials dωl have the form dωl = C(λ−3/2 + . . . )dλ (see [S, KK]), then ∞ A(Pj ) ≤ C1 λ−3/2 dλ ≤ C1 j−1 uniformly in r ∈ R1 + δ. (6.17) j 2 /4
Therefore for all P , U , z as in (6.11) and for r from R1 + δ the function f is close to one: f (U ; r, z; p) − 1 ≤ Cj−1 .
(6.18)
Using (6.12), (6.18) and the Cauchy estimate we ﬁnd that the functions Ψj (r, z) deﬁned in (6.14) are close to complex exponents: Ψj (r, z)(x) =
1 ijx e (1 + ζj (r, z)(x)) , 2π
(6.19)
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where ζj (r, z)(x) ≤ Cj −1 for r ∈ R1 + δ ⊂ Cn ,  Im z ≤ δ and  Im x ≤ δ with some jindependent δ and C = C(δ). To check the property 2a) from Deﬁnition 3 with ∆ = 1 we shall show that the map aj ζj (x) aj eijx → r 1 is 1smoothing, i.e., it sends a space H√ H0r+1 (S 1 ). To do it 0 (S ) to the space√ r −1 ijx we observe that in the Hilbert bases {( 2π j ) e }, {( 2π j r+1 )−1 eijx } of the two spaces the map has the matrix M with the entries Mlj = lr+1 j −r ei(j−l)x ζj (x) dx. Since for  Im x < δ the function ζj is analytic and bounded by Cj −1 , then mlj  ≤ Cδ (l/j)r+1 e−δj−l . Therefore the l1 norm of any row and any column of the matrix M is bounded by a constant C . Hence, a norm of the map (6.19) as a map from H0r to H0r+1 is bounded by the same C due to the Schur criterion13 and 2a) follows. The property 2b) with κ = 3 easily follows from (6.19). $ = 1 is an immediate consequence The property 2c) with dA +dJ = 3 and ∆ of the asymptotic (6.16).
The system satisﬁes a)–d). 14 The ﬁrst assertion of a) follows from the convergence νj (r) = −2iΩ3 (Pj ) → − 14 j 3 as r → 0, which implies that for small r all the functions νj are distinct. The second assertion follows from the item 2c) of Deﬁnition 3 which is checked already with dA + dJ = 3 and $ = 1. ∆ The assumption b) follows from 2a), 2c) since the exponents νj are real (for real r). The assumptions c), d) are now empty since all the Floquet exponents are analytic functions. Finally for the domain R as in (4.5) we have proved the following result: Theorem 4. For any γ > 0 there exists a subset R1 R, mes(R \ R1 ) < γ, such that on T12n = Φ0 (R1 × Tn ) ⊂ T 2n the system of Floquet solutions (6.14) with j ∈ ZV is complete nonresonant (in any space H0d , d ≥ 1). 6.3
Higher KdVequations
The lth equation from the KdVhierarchy has an [L, A]pair with the same Loperator L = −∂ 2/∂x2 − u and with some Aoperator of the form A = Al = const ∂ 2l+1/∂x2l+1 + . . . (see [DMN,MT,ZM]). Solutions χl of equation (6.1) with A = Al are given by the Its–Matveev formula (6.6), where the diﬀerential Ω3 should be replaced by an appropriate diﬀerential Ω2l+1 and 13
14
The criterion states that a norm of a linear operator which in Hilbert bases of the two Hilbert spaces has a matrix {Mlj }, is no bigger than (supl j Mlj  · supj l Mlj )1/2 . See [HS] or [KK]. see (5.4) and below.
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the vector W – by the vector W l , formed by bperiods of Ω2l+1 . We get Floquet solutions vjl of the linearised lth equation, l
vjl (t, x; r, z) = eiνj (r)t Ψj (r, z0 + W l t)(x) ,
j ∈ ZV ,
where νjl = 2Ω2l+1 (Pj ) and Ψj is given by (6.14). Similar to the KdVcase, we ﬁnd that νjl = 2(i/2)2p+1 j 2l+1 + O(j 2l−3 ),
j ∈ NV .
(6.20)
The system of Floquet solutions {vjl } is complete. Indeed, the items of Deﬁnition 3 from 1) through 2b) describe properties of the sections Ψj which are the same as for the KdV equation, so we have already checked them. The $ = −∆ $ l = 2l − 3 follows from (6.20). property 2c) with −∆ The linearised lth equation satisﬁes the assumption v): its ﬂowmaps Sτt ∗ are welldeﬁned linear isomorphisms. Indeed, by Lemma 8, outside the singular set Rs × Tn the vectors {Ψj (r, z)} form an equivalent complex basis of the skewnormal space Tu⊥c T 2n ⊂ Zd , where d ≥ 1 and u = Φ0 (r, z). After we choose these bases in the spaces Tu⊥c T 2n and Tu⊥c T 2n , the 0 (τ ) 0 (t) l
map Sτt ∗ becomes diagonal with the unit diagonal elements {eiνj (r)(t−τ ) }. So Sτt ∗ (u0 (τ ))d,d ≤ C(r, d) for all τ and t, if d ≥ 1 and r ∈ / Rs . 6.4
Linearised SG equation
Let us consider the SG equation, linearised about any ﬁnitegap solution u(t, x) as in (4.7), (4.9). Now it is more convenient to study the SG equation in the form (3.3). Accordingly, the linearised equation takes the form: ¨˜ = u u ˜xx − (cos u(t, x))˜ u,
˜˙ . w ˜ = −A1/2 u
(6.21)
As in the KdVcase, solutions of equations (6.1), i.e. of the nonautonomous Aequation and its adjoint, are given by explicit thetaformulas (see [EFM, BT]). Accordingly, Theorem 3 provides us with explicit formulas for solutions of equation (6.21) which satisfy the periodic/antiperiodic boundary conditions. Similar to the KdVcase, these solutions have the Floquet form: ˜ u ˜j (t, x; D, r) = G(V˜ x + W t + D; Pj )e(Ω1 +Ω2 )(Pj )W +(Ω1 −Ω2 )(Pj )t , ˜˙ j ; j ∈ Zn . w ˜j = −A−1/2 u (6.22)
Here G(q; p) is some analytic function of q ∈ Tn and P ∈ Γ, expressed via the θfunction, and Ω1 (P ), Ω2 (P ) are Abel integrals on Γ. The points Pj are deﬁned as solutions for the following equation: 1 (Ω1 + Ω2 )(Pj ) = j, i
j ∈ ZV ,
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(cf. (6.12) and [EFM]). Similar to the KdVcase, the Floquet exponents νj (r) = 1i (Ω1 − Ω2 )(Pj ) can be described as the last components of frequency ˜ ∈ Rn+1 corresponding to surfaces T 2n+2 , where except the previvectors W ¯1 ], [E −1 , E ¯ −1 ], . . . , [E −1 , E ¯ −1 ] we also εopen closed ously opened gaps [E1 , E n n 1 1 gaps at the points Pj and 1/Pj , and next sent ε to zero. In striking diﬀerence with the KdV case, some exponents νj are nonreal if the open gap are suﬃciently large; corresponding functions r → νj (r) have algebraic singularities, see [McK, EFM]. Asymptotical analysis of solutions (6.22) show that after multiplying them by proper constants the solutions take the form u ˜j =
1 i(jx+νj (r)t) e (1 + ζj (r, D)(x)) , w ˜j = . . . , 2π
(6.23)
where the function ζj goes to zero when j → ∞ or r → 0. Due to the same arguments as in the KdV case, the system of Floquet solutions (6.23) satisﬁes the asymptotic assumption 2) from Deﬁnition 3, and due to (4.10) it is nonresonant (see [BiK1, BoK2, KK]). Using the Corollary from Lemma 10, we get that this system of Floquet solutions of the linearised equation (SG)+(OP) is complete nonresonant.
7 7.1
Normal form A normal form theorem
We continue to study the Hamiltonian equation (5.1) near an invariant manifold T 2n = Φ0 (R × Tn ) which possesses the properties (i)–(v) as in Section 3. Proposition 2 puts the linearised equation (5.2) to a constant coeﬃcient normal form, provided that this equation possesses a complete nonresonant system of Floquet solutions. In this section we show that under these assumptions the equation (5.1) itself can be put to a convenient normal form in a neighbourhood of T 2n . Namely, we show that the actionangle variables (p, q) on T 2n can be supplemented by a skeworthogonal to T 2n vectorcoordinate y in such a way that in the new coordinates the equation is Hamiltonian with a hamiltonian h(p) + 12 B(p)y, y + h3 (p, q, y),
h3 = O(y3 ) .
Here B(p) is the selfadjoint operator from Proposition 2 and the term h3 deﬁnes a hamiltonian vector ﬁeld of the same order as the nonlinear part J∇H of the original equation (this is a crucial property of the normal form!). The normal form is an eﬀective technical tool to study equation (5.1) and its perturbations in the vicinity of T 2n . We assume that the linearised equation (5.2) has a complete family of skeworthogonal Floquet solutions vj (t) as in (5.4), deﬁne the singular subset Rs , Rs = Rsc ∩ R as in Section 5 (see there a remark after Deﬁnition 3). As
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in Section 5, we choose any domain R1 in a compact part of the regular set R0 = R \ R s . By Lemma 5 the equation (3.1) is integrable in Φ0 (R0 × Tn ). So we can cover Φ0 (R1 × Tn ) by a ﬁnite system of open subdomains such that in each one the equation admits analytic actionangle variables (p, q). For simplicity we suppose that the actionangles exist globally in Φ0 (R1 ×Tn ) and use them instead of (r, z). Accordingly, we write Φ0 (R1 × Tn ) as Φ0 (P × Tn ), where P = {p} Rn and Tn = {q}. We denote W = P × Tn and Φ0 (W ) = T12n . The map Φ0 W → T 2n ⊂ Z analytically extends to a bounded analytic map W c → Z c , where W c is a complex neighbourhood of W , W c = (P + δ) × {q ∈ Cn/2πZn   Im q < δ}. Since ω = ∇h (see Lemma 5), then we write the skeworthogonal Floquet solutions vj (t) as vj (t; p, q) = eiνj (p)t Ψj (p, q + t∇h(p)) ,
p ∈ P, q ∈ Tn , j ∈ Zn .
(7.1)
The linear in y map y → Φ1 (p, q + t∇h)y as in (5.8) reduces the linearised equation (5.2) to the constantcoeﬃcient linear equation y˙ = JB(p)y, . . . ,
(7.2)
where the dots stand for components of the linearised equation in directions tangent to W × {0} (see Proposition 2). We denote by Sδ = Sδ (Yd ) the manifold Sδ = W × Oδ (Y ), Y = Yd , and denote by Sδc its complex neighbourhood Sδc = W c × Oδ (Y c ). We give Sδ symplectic structure by means of the 2form (dp∧dq)⊕α2Y , where α2Y = α2 Y . Our goal in this section is to prove the following Normal Form Theorem: Theorem 5. Let the Hamiltonian equation (5.1) and its invariant submanifold T 2n satisfy the assumptions i)–v); let a subdomain T12n = Φ0 (P ×Tn ) ⊂ T02n be as above and (7.1) be a complete system of skeworthogonal Floquet solutions of the linearised equation (5.2). Then there exists δ1 > 0 and an analytic symplectomorphism G (Sδ1 , dp∧dq ⊕α2Y ) → (Z, α2 ) such that G(Sδ1 ) is a neighbourhood of T12n and H ◦ G = h(p) + 12 B(p)y, y + h3 (p, q, y). Here h3 = O(y3 ) is an analytic functional such that its gradient map is of $ − dJ }, i.e. ∇y h3 (p, q, y) ˜ ≤ Cy2 for order d˜ = max{dH , −∆ − dJ , −∆ s s−d any (p, q, y) ∈ Sδ1 . To simplify our presentation we suppose below that all the frequencies νj (p) are real and consequently the operator B(p) is diagonal in the ϕj basis ν (p) of the space Y , i.e., B(p)ϕj = jν J ϕj for every j. j
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We start with the aﬃne in y ∈ Y map Φ, Φ = Φ0 + Φ1 Sδc → Z c ,
(p, q, y) → Φ0 (p, q) + Φ1 (p, q)y.
It is real (it sends Sδ to Z), bounded on bounded subsets of Sδc and is weakly analytic by assumptions b) and d). So Φ is an analytic map by the criterion of analyticity. By Lemma 8 its linearizations at points from W × {0} deﬁne isomorphisms of R2n × Y and Z. Thus, by the inverse function theorem the map Φ deﬁnes an analytic isomorphism of Sδc and a complex neighbourhood of T12n in Z, provided that δ is suﬃciently small Next we study symplectic properties of the map Φ. Since restriction of Φ to W × {0} equals Φ0 and restriction to any disc {w} × Oδ (Y ) equals Φ1 (w) up to a translation, then these restrictions are symplectic. In particular, for any w ∈ W the map Φ∗ (w, 0) is a linear symplectomorphism. Hence, the pullback form ω2 := Φ∗ α2 equals (dp ∧ dq) ⊕ α2Y for w = 0 and these two forms coincide being restricted to any disc {w} × Oδ (Y ). It means that the diﬀerence ω∆ = ω2 − dp ∧ dq ⊕ α2Y may be written as ω∆ = jW W (w, y)dw ∧ dw + jW Y (w, y)dy ∧ dw + jY W (w, y)dw ∧ dy, where the linear operators jW W (w, y), jW Y (w, y) and jY W (w, y) vanish for y = 0. In the calculations we carry out below it is convenient to adopt gradientnotations for linearizations of the maps Φ and Φ1 in w. Namely, below we write Φ(w, y)∗ (δw, 0) = ∇wj Φ(w, y)δwj =: ∇w Φ · δw. Here ∇w Φ = (∇p Φ, ∇q Φ) ∈ Z × · · · × Z (2n times). Similar we write Φ1∗ (δw, 0) = ∇w Φ1 · δw, where any ∇wj Φ1 is a linear operator Y → Z. In these notations we have: ¯ 1 δy, ∇w Φ1 y · δw ω2 [δy, δw] = α2 [Φ1 δy, Φ0∗ δw + (∇w Φ1 · δw)y] = JΦ ¯ w Φ1 · δw)y, Φ1 δy. Hence, and ω2 [δw, δy] = J(∇ jW Y (w, y)δy = JΦ1 (w)δy, ∇w Φ1 (w)y, jY W (w, y)δw = Φ∗1 (w)J(∇w Φ1 · δw)y.
(7.3)
Abbreviating (δw, δy) ∈ R2n × Y to δz, we write the form ω∆ as ω∆ = J ∆ dz ∧ dz, where J ∆ is the operator matrix: ( ' jW W jW Y . (7.4) J ∆ = J ∆ (w, y) = ∗ −jW 0 Y The form ω∆ is exact, as well as the forms α2 and dp∧dq ⊕w2Y , i.e. ω∆ = dω1 . Lemma 2 represents the 1form ω1 as 1 ) * ω1 (w, y) = JΦ1 (w)y, ∇w Φ1 (w)ty dt dw =
0 1 2 JΦ1 (w)y, ∇w Φ1 (w)ydw
= 12 α2 [Φ1 (w)y, ∇w Φ1 (w)y]dw.
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We have seen that ω2 = Φ∗ α2 = (dp ∧ dq) ⊕ α2Y + d(L(w, y)dw),
(7.5)
where the 2nvector L has the components Lj = 12 α2 [Φ1 (w)y, ∇wj Φ1 (w)y]. Next we calculate how the map Φ changes the hamiltonian H. To begin with we analyse how Φ1 transforms the quadratic part Au, u of the hamiltonian H. Since the nonautonomous symplectic linear map Φt := Φ1 (p, q + t∇h) sends solutions y(t) of equation (7.2) to solutions v(t) = Φt y(t) of (5.2), then we have the equalities: v˙ + + +
Φ˙ t y + Φt y˙ + + +
JAt v + + +
Φ˙ t y + Φt JB(p)y
JAt Φt y Thus, JAt Φt y = Φ˙ t y + Φt JB(p)y.
(7.6)
Taking skewproduct of (7.6) with −v, we get: JAt Φt y, Jv + + + ∗
Φt At Φt y, y
Φ˙ t y + Φt JB(p)y, Jv + + +
(7.7)
Φ˙ t y, JΦt y + B(p)y, y,
¯ t y = JBy, Jy ¯ = By, y by symplecticity of where we use that Φt JBy, JΦ t the map Φ . Since for t = 0 we have At = A + ∇H(Φ0 (w))∗ and Φ˙ t = ∇q Φ1 (w) · ∇h(p), then relation (7.6) with t = 0 implies that Φ1 (w)JB(p) = J(A + ∇H(Φ0 (w))∗ )Φ1 (w) − ∇q Φ1 (w) · ∇h(p). Similar, (7.7) implies that ) * B(p) − Φ1 (w)∗ (A + ∇H(Φ0 (w))∗ )Φ1 (w) y, y = Φ1 (w)∗ J ∇q Φ1 (w) · ∇h(p) y, y = A(w)y, y, ¯ q Φ1 · ∇h), i.e., where A stands for the symmetrisation of the operator Φ∗1 J(∇ ¯ q Φ1 (w) · ∇h(p)) − (∇q Φ1 (w)∗ · ∇h(p))JΦ ¯ 1 (w) . A(w) = 12 Φ∗1 (w)J(∇ Since this relation holds identically in y ∈ Y , then B(p) − Φ1 (w)∗ (A + ∇H(Φ0 (w))∗ )Φ1 (w) = A(w).
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Lemma 11. The operator A deﬁnes a (∆ + dJ )smoothing symmetric map c A : Ydc → Yd+d , analytic in w ∈ W c . J +∆ Proof. The operator A is symmetric by its construction. It remains to check its smoothness. Since ∇q Φ1 = ∇q (Φ1 − ι), then by (5.10) and the Cauchy estimate the operator ∇q Φ1 · ∇h analytically depends on w ∈ W c as a map Ydc → Zd+∆ . c c By Lemma 9 the operator Φ1 (w)∗ J¯ : Zd+∆ → Yd+d also is analytic in w. J +∆ Hence, the ﬁrst term of the operator A deﬁnes an analytic in w ∈ W c map c Ydc → Yd+d . J +∆ ¯ 1 (w) : Y c → Z c Using Lemma 9 once again we ﬁnd that the operator JΦ d d+dJ is analytic in w. Due to the second assertion of this lemma and the Cauchy c c estimate, the map ∇q Φ∗1 · h : Zd+d → Yd+d is analytic in w as well. J J +∆ Combining these two statements, we ﬁnd that the second term of A also c deﬁnes an analytic in w ∈ W c map Ydc → Yd+d . This completes the J +∆ proof. Now we write the transformed hamiltonian as H ◦ Φ = 12 AΦ0 , Φ0 + AΦ0 , Φ1 y + 12 AΦ1 y, Φ1 y + H(Φ), and separate its aﬃne in y part: H◦Φ=
1 2 AΦ0 , Φ0
+
! ! + H(Φ0 ) + AΦ0 , Φ1 y + ∇H(Φ0 ), Φ1 y
1 2 AΦ1 y, Φ1 y
! + H(Φ0 + Φ1 y) − H(Φ0 ) − ∇H(Φ0 ), Φ1 y .
The ﬁrst term in the righthand side equals h(p). By Lemma 2 the form ω2 = Φ∗ α2 equals (dp ∧ dq) ⊕ α2Y , when y = 0. Hence, for y = 0 the ycomponent of equation (5.1), written in the (p, q, y)variables, is J∇y (H ◦ Φ). It equals zero since the set {y = 0} is invariant for the equation. Thus, the second term vanishes. By Lemma 11, AΦ1 y, Φ1 y = By, y − ∇H∗ Φ1 y, Φ1 y − Ay, y . Therefore the third term in the r.h.s. equals 12 B(p)y, y + h2 (p, q, y), where h2 = − 12 ∇H(Φ0 )∗ Φ1 y, Φ1 y − 12 A(w)y, y + H(Φ0 + Φ1 y) − H(Φ0 ) − ∇H(Φ0 ), Φ1 y. It is easy to see, using Lemmas 9 and 11, that h2 deﬁnes an analytic gradient map ∇y h2 R2n × Yd → Yd−d˜. Thus, the aﬃne in y map Φ transforms the hamiltonian H to H ◦ Φ = h(p) + 12 B(p)y, y + h2 (p, q, y), ˜ where h2 = Oy2 and ord ∇h2 = d. Our next goal is to normalise the symplectic structure ω2 = Φ∗ α2 in Sδ by means of the Moser–Weinstein theorem (Lemma 3). The theorem states that
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ϕ∗ ω2 = (dp ∧ dq) ⊕ α2Y , where ϕ is the timeone shift S01 along trajectories of a nonautonomous equation: z˙ = V t (z) , z = (w, y). The vector ﬁeld V t Sδ → R2n × Yd is obtained as a solution of the equation −(J 0 + tJ ∆ )V t = a(z, y), where J 0 (δp, δq, δy) = (−δq, δp, Jδy), the operator J ∆ is as in (7.4) and the map a is such that diﬀerential of the 1form a(z)dz equals ω2 − (dp ∧ dq) ⊕ α2Y . By (7.5), a(z) = (L(z), 0). We claim that the map ϕ sends Sδc1 to Sδc (δ1 is suﬃciently small compare to δ) and transforms H ◦ Φ to a hamiltonian of similar form: Lemma 12. The hamiltonian H ◦ Φ ◦ ϕ equals to H ◦ Φ ◦ ϕ = h(p) + 12 B(p)y, y + 12 B(p, q)y, y + h3 (p, q, y),
(7.8)
where B(p) is the same as in (7.2) and B(p, q) is a linear operator of or˜ analytic in (p, q) (d˜ is the same as in Theorem 5)). The function der d, ˜ ∇y h3 (p, q, y) ˜ ≤ h3 = O(y3 ) has an analytic gradient map of order d, d+d 2 Cy . The statement of the lemma is quite obvious for a ﬁnitedimensional phase space Sδ , but not in the inﬁnitedimensional situation. Indeed, the transformation ϕ has the form ϕ = id +ϕ, $ where ϕ $ = y2 is a ∆smoothing map. Thus the transformed hamiltonian gets the term B(p)y, ϕy $ which is O(y3 ) with the gradient of order dA − ∆. The number dA − ∆ could be rather big and the term could spoil the forthcoming constructions. Fortunately, it vanishes up to a smoother term. This is essentially what the lemma states. For its proof see Lemma 7 in [K1] and [KK]. To prove the theorem it remains to check that the operator B in (7.8) vanishes. Since ϕ is analytic and O(y2 )close to the identity, then ϕ∗ (w, 0){0}×Y = id . Thus the transformation y(t) → (Φ ◦ ϕ)∗ (p, q + t∇h, 0) y(t) sends solutions of the equation (7.2) to solutions of (5.2). From other side, Φ ◦ ϕ is a canonical transformation which transforms solutions of the equation with hamiltonian (7.8) to solutions of (5.1). In particular, it sends the curves w(t;p, q) = (p, q + t∇h(p), 0) to solutions u0 (t) of (5.1). Hence, the linearisation Φ ◦ ϕ(w(t)) ∗ transforms solutions of the linearised equation y˙ = J B(p) + B(w(t) y, . . . (7.9) to solutions of (5.2). Therefore the transformation ϕ(w(t))∗ sends solutions of (7.9) to solutions of (7.2). Since a ycomponent of the map ϕ(w(t))∗ is the identity, then we must have JB(p)y ≡ J(B(p) + B(p, q + t∇h))y. This implies that B ≡ 0 and the theorem is proven.
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Examples
1)Korteweg–de Vries equation. The KdV equation in a Sobolev space Zs = H0s (S 1 ) with s ≥ 3, which is given symplectic structure by the form α2 = (−∂/∂x)−1 du, duL2 takes the form (3.1) (see Example 3.1). Its restriction to a bounded part T 2n of any ﬁnitegap manifold TV2n (see (4.5)) satisﬁes the restrictions i)–v) and the corresponding system of Floquet solutions (6.13) $ = 1, dJ = 1, dH = 0 and dA = 2. is complete nonresonant with ∆ = ∆ Therefore Theorem 5 provides KdV with a normal form. To state the result, we ﬁnd the singular subset Rs ⊂ R 15 and choose any domain R1 R \ Rs . We cover R1 up to its zeromeasure subset by nonoverlapping subdomains R11 , R12 , . . . such that the KdVequation restricted to any manifold Φ0 (R1j × Tn ) = Jj2n admits actionangle variables (p, q) with p ∈ Pj Rn . For any s we denote by Ys ⊂ H0s (S 1 ) the closed subspace spanned by the functions {cos jx, − sin jx  j ∈ NV }. Applying Theorem 5 we get: Theorem 11. For any s ≥ 3 there exists δ > 0 and an analytic symplectomorphism G of (Pj ×Tn ×{ys < δ}, dp∧dq⊕α2Y ) and a domain in (H0s , α2 ). It contains Tj2n in its range and G−1 transforms KdV to the Hamiltonian system q˙ = ∇p H,
p˙ = −∇q H,
y˙ =
∂ ∇y H ∂x
(7.10)
with a hamiltonian H of the form H = h(p) + 12 B(p)y, y + H3 (p, q, y). Here h(p) is the hamiltonian of KdV restricted to Tj2n , B(p) is the linear operator in Yd with eigenvectors cos mx, − sin mx and eigenvalues νm (p) (m ∈ NV ), H3 = O(y3s ) is a function with a zeroorder analytic gradient map. 2) Higher KdV equations. Let us take any lth equation from the KdVhierarchy. Since the same (as in the KdVcase) sections Ψj of the skewnormal bundle to a ﬁnitegap manifold TV2n give rise to its Floquet solutions, then the same map Φ1 reduces the linearised lth equation to the equation y˙ = JB l (p)y in the space Y . Here J = ∂/∂x and B l (p) is a linear operator with the eigenvectors cos jx, − sin jx, corresponding to the eigenvalues νjl (p) (j ∈ NV ) as in (6.20) Therefore, the same map G with s ≥ 2p + 1 reduces the lth KdV equation in the vicinity of Tj2n (the same as above part of TV2n ) to the equation (7.10) with H = Hl (p, q, y) = hl (p) + 12 B l (p)y, y + H3l (p, q, y). Here hl is the hamiltonian of the lth equation restricted to Tj2n and the operator B l (p) has the eigenvalues νjl . Now ∆ = dJ = 1 as in the KdVcase, dA = 2l, $ = 3 − 2l by (6.20). So d˜ = 2l − 2 and H3 = O(y3 ) has dH = 2l − 2 and ∆ s an analytic gradient map of order 2l − 2. 3) SineGordon equation. 15
In the KdVcase the set Rs is empty. We neglect this nice speciﬁcity of KdV since our goal is to present arguments applicable to other integrable PDEs.
KAMpersistence of ﬁnitegap solutions
8 8.1
111
The KAM theorem The main theorem and related results
Let ({Zs }, α2 ), α2 = J dz ∧ dz be a scale of symplectic Hilbert spaces and ord J¯ = −dJ ≤ 0. Let H be a quasilinear hamiltonian of the form H = 1 2 Az, z + H(z), where A is a selfadjoint isomorphism of the scale of order dA > −dJ . We ﬁx any d ≥ dA /2 and assume that the function H is analytic in the space Zd (or in a neighbourhood in Zd of the manifold T02n , see below) and deﬁnes an analytic gradient map of order dH , ∇H : Zd → Zd−dH . We have dH < dA due to the quasilinearity of the hamiltonian H. The corresponding Hamiltonian equation takes the form: u˙ = J∇H(u) = J(Au + ∇H(u)),
J = (−J)−1 .
(8.1)
As in Sections 3 and 5 we assume that the equation (8.1) has an invariant manifold T02n = Φ0 (R0 × Tn ) ﬁlled with quasiperiodic solutions u0 (t; r, z) which satisﬁes the assumptions i)–v). The manifold R0 is the regular part $ we denote any chart on R0 of an ndimensional real analytic set R. By R analytically diﬀeomorphic to a bounded connected subdomain of Rn . We $ with this domain and supply it with the ndimensional Lebesgue identify R measure mesn . As in Section 5, we also consider linearisation of the equation (8.1) about a solution u0 : (8.2) v˙ = J Av + ∇H(u0 (t))∗ v , and assume that (8.2) has a system (5.4) of Floquet solutions vj (t). Our concern in this section is a hamiltonian perturbation of the equation (8.1): u˙ = J(Au + ∇H(u) + ε∇H1 (u)),
(8.3)
and behaviour of solution for (8.3) near the manifold T02n . We assume that H1 is an analytic functional such that ord ∇H1 = dH . By d˜ we denote the real number from Theorem 5, d˜ = max {dH , −∆ − $ dJ , −∆−d J }, where −∆ is the order of the linear operator Φ1 −ι (see (5.10)) $ is the exponent of growth in j of “variable parts” of the the Floquet and ∆ exponents νj (r) (see (5.12)). Let us ﬁx any ρ˜ such that 0 < ρ˜ < 1/3. Now we state a KAM theorem which is the main result of this paper: Theorem 7 (the Main Theorem). Let the invariant manifold T02n satisfy the assumptions i)–v) and the system of Floquet solutions (5.4) for (8.3) is complete nonresonant. Besides, d1 := dA + dJ ≥ 1 and 1 2 1) (spectral asymptotic): νj (r) = K1 j d1 + K11 j d1 + K12 j d1 + · · · + ν˜j (r), where K1 > 0, d1 > d11 > . . . (the dots stand for a ﬁnite sum), the functions ν˜j analytically extend to Rc , where ˜ νj  ≤ Cj κ with some κ < d1 − 1;
112
S. B. Kuksin
2) (quasilinearity): d˜ < d1 − 1. Then most of the invariant tori T n (r) of equation (8.1) persist in (8.3) $ ⊂ R0 as above and any when ε → 0 in the following sense: for any chart R $ε R $ and a Lipschitz embedding suﬃciently small ε > 0, a Borel subset R $ε × Tn −→ Zd can be found such that: Σε : R $\R $ε ) −→ 0 as ε −→ 0, a) mesn (R $ε × Tn −→ Zd is bounded by Cερ˜ as well as its b) the map (Σε − Φ0 ) : R Lipschitz constant and is analytic in q ∈ Tn ; $ε , is invariant for the equac) each torus Tεn (r) := Σε ({r} × Tn ), r ∈ R tion (8.3) and is ﬁlled with its timequasiperiodic solutions hε (t) of the form hε (t) = hε (t; r, z) = Σε (r, z + tωε (r)), where ωε − ω + Lip (ωε − ω) ≤ Cερ˜. Ampliﬁcation. The statements b), c) of Theorem 7 remain true with ρ˜ replaced by any ρ < 1. Besides, Σε − Φ0 d + ωε − ω ≤ Cε. n , ), W , = R×T $ ,ε ), W ,ε = R $ε ×Tn . and T$ε2n = Σε (W We denote T$ 2n = Φ0 (W 2n 2n The set T$ε is a remnant of the invariant manifold T$ in the perturbed equation (8.3). Since T$ 2n is a 2ndimensional manifold embedded to Zd , then its 2n$ 2n is ﬁnite and positive. The remnant dimensional Hausdorﬀ measure mesH 2n T 2n $ set Tε is very irregular (it is totally disconnected). Still it carries most of a measure of the set T$ 2n :
$ 2n is bigger Proposition 4. Under the assumptions of Theorem 7, mesH 2n Tε than mesH T$ 2n − o(1) as ε → 0. 2n
Proof.
16
By the assertion a) of the theorem, , , mesH 2n (W \ Wε ) = o(1).
(8.4)
,ε →−→ Φ0 (W ,ε ) ⊂ T$ 2n is Lipschitz and has a LipsThe map Φ0 : W ∼
,ε ) →−→ T$ 2n has the form : Φ0 (W chitz inverse, so the map Σε ◦ Φ−1 ε 0 ∼
$ 2n ≥ id + L, where Lip L ≤ Cερ˜ (we use the assertion b)). Therefore, mesH 2n Tε H ρ ˜ H ,ε ) − O(ε ). Since mes (Φ0 (W \ W ,ε )) = o(1) by (8.4), then the mes2n Φ0 (W 2n assertion follows. Under the assumptions of Theorem 7, a solution u0 (t; r, z) of (8.1) is linearly stable if all Floquet exponents νj (r) are real (see the Corollary to Propo$ Then the solutions sition 2). Let us assume that this is the case for all r ∈ R. $ε also are linearly stahε (t; r, z) of the perturbed equation (8.3) with r ∈ R ble, provided that this equation linearised about hε satisﬁes some a priori estimate. We recall that by the assumption v) the ﬂow maps Sτt ∗ (hε (τ )) of 16
For basic properties of the Hausdorﬀ measure which we use below see e.g. [Fal].
KAMpersistence of ﬁnitegap solutions
113
the linearised equation are well deﬁned in the space Zd . We say that the linearised equation is uniformly well deﬁned (in Zd ) if Sτt ∗ (hε (τ ))d,d ≤ C1 eC2 t
for all t, τ.
(8.5)
Theorem 8. If under the assumptions of Theorem 7 all the Floquet expo$ then a solution hε (t) is linearly stable, nents νj (r) are real for all r ∈ R, provided that equation (8.3) linearised about this solution is uniformly well deﬁned. (Examples we consider below in Section 9 show that the assumption (8.5) is quite nonrestrictive). We prove the two theorems and the ampliﬁcation, reducing them to similar statements concerning perturbations of parameterdepending linear systems. 8.2
Reduction to a parameterdepending case
We perform the reduction in four steps. Step 1 (localisation). Let us denote by Rf the set of singularities of the $  det ω∗ (r) = 0}, and denote R $s = (Rs ∩ R) $ ∪ frequency map ω, Rf = {r ∈ R Rf , where Rs is the singular set, constructed in Section 5. By the assumption $ So R $s also is one, and for any iv), Rf is a proper analytic subset of R. given positive γ0 we can ﬁnd a ﬁnite system of M connected subdomains $l ⊂ R $\R $s such that dist (rj , rj ) ≥ C(γ0 ) > 0 if rj ∈ R $j and rj ∈ R $j with R j = j. Besides, ˜ \ ∪R $l ) < γ0 , a) mes (R $l ×Tn ) admits analytic actionb) the hamiltonian system restricted to Φ0 (R n angle variables (p, q), where p ∈ Pl R and q ∈ Tn . The map (p, q) → (r, z) has the form r = r(p), z = q+z0 (p). This map, its inverse and the hamiltonian h = hl (p) all are δanalytic with some positive δ = δ(γ0 ). By Lemma 2.2, ∇h(p) ≡ ω(r(p)); c) for every l the gradient map p → ∇h(p) ≡ ω(r(p)) deﬁnes a diﬀeomorphism Pl →−→ Ωl Rn which is δanalytic as well as its inverse; ∼
$l is connected, then the eigenvectors ψj of the d) since each domain R $l . operator JB(r) are rindependent when r ∈ R Step 2 (a normal form theorem). In this step of the proof and in the next $l as above and drop the index l. Step 3 we consider any ﬁxed domain R Applying Theorem 5 we ﬁnd an analytic symplectomorphism G which $ × Tn ) to the form transforms the equation (8.1) in the vicinity of Φ0 (R given in the theorem. The same symplectomorphism converts the perturbed equation (8.3) to the Hamiltonian system p˙ = −∇q Hε , q˙ = ∇p Hε , y˙ = J∇y Hε .
(8.6)
Here p ∈ P , q ∈ Tn , y ∈ Oδ (Yd ) and Hε = h(p) + 12 B(p)y, y + h3 (p, q, y) + ˜ The operator B(p), the εH1 (p, q, y) with h3 = O(y3d ) and ord ∇y h3 = d.
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functions h, h3 , H1 and their gradients all are δanalytic in the corresponding domains. Step 3 (introducing a parameter). Let us consider the following neighbourhoods of the torus T0n = {0} × Tn × {0} in Y and Y c : Qδ = Oδ (Rn ) × Tn × Oδ (Yd ) ⊂ Y, Qcδ = Oδ (Cn ) × {Im q < δ} × Oδ (Ydc ) ⊂ Y c . In the equation (8.6) we perform a shift of the action p: p, q˜, y˜), (p, q, y) = (˜ p + a, q˜, y˜) =: Shifta (˜ where a ∈ P is a parameter of the shift. After this transformation hamiltonian p, q˜, y˜; a) of the tildevariables from the Hε becomes an analytic function Hε (˜ domain Qcδ . It has the following form: ˜ 3 (˜ y , y˜ + εH1 (˜ p + a, q˜, y˜) + h p, q˜, y˜; a), Hε = h(a) + ∇h(a) · p˜ + 12 B(a)˜ ˜ 3 ˜ = O(˜ ˜ 3 = O(˜ y 3d + ˜ p2 + ˜ p˜ y 2d ) and ∇y h y 2d + ˜ p˜ y d ) (so where h d−d ˜ ˜ ord ∇h3 = d). ˜ 3 and the Floquet exponents νj are analytic bounded The functions h, H1 , h functions of the parameter a ∈ P +δ. Because the property c) from Step 1, the map P a → ω = ∇h(a) ∈ Ω deﬁnes an analytic Lipschitz diﬀeomorphism of P and a bounded domain Ω ⊂ R. We drop the tildes and change the parameter a to ω. Now the hamiltonian Hε reeds as Hε (p, q, y; ω) = h(a) + ω · p + 12 B(ω)y, y + εH1 (p, q, y; ω) + h3 (p, q, y; ω). The operator JB is diagonal in the symplectic basis {ψj }, constructed in Proposition 2: Jψj = ±iνjJ ψj , JBψj = νj (r)ψj
∀ j ∈ Zn .
Since the hamiltonian Hε is δanalytic, then by the Cauchy estimate it is Lipschitz in a ∈ P as well as in ω ∈ Ω. This is all we need from its dependence in the parameters. In the vicinity of the torus T0n = {0} × Tn × {0} in Qδ the hamiltonian Hε is a perturbation of the qindependent hamiltonian H0 = ω · p + 12 B(ω)y, y (modulo the irrelevant constant h(a)). Indeed, ε is small and the term h3 has on T0n a highorder zero. The hamiltonian equations with the hamiltonian Hε (p, q, y; ω) take the form: p˙ = −∇q (εH1 +h3 ), q˙ = ω + ∇p (εH1 + h3 ), y˙ = J(B(ω)y + ε∇y H1 + ∇y h3 ).
(8.7)
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115
We abbreviate (p, q, y) to h and rewrite (8.7) as h˙ = VHε (h). In the context of equations (8.7), we call the functions νj (ω) frequencies of the linear equation. Hamiltonian vector ﬁelds with hamiltonians of the form Hε are studied in [K, K2, P]. Now we break the proof of Theorem 7 to present the main result of these works. After this we make the last step to complete the proof of Theorem 7. 8.3
A KAMtheorem for parameterdepending equations
To state the theorem we need, we relax restrictions on the hamiltonian Hε as in the assumptions 1)–3) below: 1) (frequencies). The complex functions νj (ω), j ∈ Zn , are Lipschitz, are real for j ≥ j1 with some j1 ≥ n + 1 and are odd in j, νj ≡ −ν−j . For j ≥ n + 1 and for some ﬁxed ω0 ∈ Ω the following estimates hold: 1
2
˜
˜
νj (ω0 ) − K1 j d1 − K11 j d1 − K12 j d1 − . . .  ≤ Kj d and Lip νj ≤ Kj d , where K1 > 0, d1 ≥ 1, d˜ < d1 − 1 and the dots stand for a ﬁnite sum with some exponents d1 > d11 > d21 > . . . . 2) (perturbation). The functions h3 and H1 are analytic in (p, q, y) ∈ Qcδ and everywhere in Qcδ satisfy the estimates: H1  + ∇y H1 d−d+d ˜ J ≤ 1 ∀ω, 2 2 3 h3  ≤ K(p + pyd + yd ) ∀ω, 2 (8.8) ∇y h3 d−d+d ˜ J ≤ K(pyd + yd ) ∀ω, the same estimates hold for Lipschitz constants in ω ∈ Ω of these functions and their gradients. 3) (domain of parameters). Ω is a bounded Borel set in Rn of positive Lebesgue measure, such that diam Ω ≤ K2 and ω ≤ K for every ω ∈ Ω. Let us choose any ρ ∈ (0, 13 ). Theorem 9. Suppose that the assumption 1)3) hold. Suppose also that there ˜ K, K1 , K2 and exist integer j2 ≥ n and M1 , depending only on n, d1 , d, K11 , K12 . . . , such that s · ω + ln+1 νn+1 (ω) + · · · + lj2 νj2 (ω) ≥ K3 > 0
(8.9)
for all ω ∈ Ω, all integer nvectors s and all j2 vectors l such that s ≤ M1 and 1 ≤ l ≤ 2. Then for arbitrary γ > 0 and for suﬃciently small ε ≤ ε¯(γ) (¯ ε > 0), a Borel subset Ωε ⊂ Ω and a Lipschitz embedding Σε : Tn × Ωε −→ Qδ can be found with the following properties: a) mes (Ω \ Ωε ) ≤ γ,
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b) the map Σε is Cερ close to the map Σ0 : (q, ω) → (0, q, 0) ∈ Qδ and Lipschitz constant of the diﬀerencemap Σε − Σ0 is bounded by Cερ as well; c) each torus Σε (Tn × {ω}), ω ∈ Ωε , is invariant for the ﬂow of equation (8.7) and is ﬁlled with its quasiperiodic solutions h(t) of the form h(t; q, ω) = Σε (q + ω t, ω), where ω = ω (ω) and ω − ω + Lip (ω − ω) ≤ Cερ . In statement b) of the theorem we view the diﬀerence Σε − Σ0 as a map, valued in the Hilbert space R2n × Yd . Ampliﬁcation. Assertions b), c) hold with ρ replaced by one. Theorem 10. If in Theorem 9 all the frequencies νj are real, then a solution h(t) is linearly stable provided that the equation (8.7) linearised about this solution is uniformly well deﬁned in the space R2n × Yd . For proofs these results with d1 ≥ 1, d˜ ≤ 0 see [K,P] and with d1 > 1 see [KK, K2]. 8.4
Completion of the Main Theorem’s proof (Step 4)
Now we apply Theorem 9 to equation (8.7) with Ω equal to a Borel subset Ωl ,l = {ω(r)  r ∈ R ,l }, l = 1, . . . , M , which we construct below, of the domain Ω and with γ = γ0 /(M C0 ), where the number C0 will be be chosen later. The assumptions 1)–3) hold with the constants from n through d11 , d21 , . . . the same as in Theorem 7, while the constants K and K2 depend on γ. We take j2 = j2 (γ) and M1 = M1 (γ) as in Theorem 9 and consider all resonances as in the l.h.s. of (8.9). Since the system of Floquet exponents {νj (r)} is nondegenerate, then each resonance does not vanish identically. As these functions are analytic, we can ﬁnd K3 = K3 (γ) and for every l can ,l such that mes (Ω $ l \ Ωl ) ≤ γ/M and (8.9) holds for all ﬁnd a subset Ωl ⊂ Ω ω ∈ Ωl . For every l we apply Theorem 9 with γ as above to ﬁnd the subset Ωlε ⊂ Ωl , mes (Ωl \ Ωlε ) < γ0 , and the map Σlε : Tn × Ωlε −→ Qδ . $ε ⊂ R $ and the map Σε : R $ε × Now we are in position to deﬁne the set R Tn −→ Qδ , claimed in Theorem 7. We set: $ε = R
M
{r ∈ Rl  ω(r) ∈ Ωlε },
Σε (r, z) = G ◦ Shiftp ◦ Σlε (q(z), ω(r))
l=1
for r in Rl , where (r, z) → (p, q) is the actionangle transformation from Step 1. $ε and the map Σε satisfy all the claims of Theorem 7. Indeed, The set R $ε ) = mes (R $ \ ∪R) ˜ + mes (R \ R
M
,l  ω(r) ∈ Ω $ l \ Ωl } mes {r ∈ R
l=1
+
M l=1
,l  ω(r) ∈ mes {r ∈ R / Ωlε }.
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117
$ε ) is bounded Denoting sup det ∂ω/∂r−1 by C(γ0 ) we see that mes (R \ R by γ0 + γ0 + C(γ0 )M γ0 /(M C0 ). This is smaller than 3γ0 , if C0 = C0 (γ0 ) is big enough. It means that we can choose γ = γ(ε) in such a way that $ ≤ 3γ0 goes to zero with ε and the assertion a) of Theorem 7 mes (R \ R) holds. The tori Σε ({r} × Tn ) are invariant for equation (8.3) and are ﬁlled with its quasiperiodic solutions of the form hε (t), where ωε = ω (p(r)). The estimates for Σε − Φ0 and ωε (r) − ω(r) readily follows from the corresponding estimates in Theorem 9. It remains to estimate Lipschitz constants of the diﬀerences as above. Let $ε × Tn . If r1 and r2 belong to us take any two points (r1 , z1 ) and (r2 , z2 ) in R $l , then the estimates for increments17 of Σε − Φ0 and ωε − ω the same set R follow from the corresponding estimates for the increments of Σ(0, q, 0) and ω −ω since the maps (8.16) are Lipschitz. If r1 and r2 belong to diﬀerent sets $l , then r1 − r2  ≥ C(γ) > 0 and the increments of the diﬀerences divided R by the increments of the arguments is bounded by C1 ερ /C(γ). Since we can choose the rate of decaying γ(ε) → 0 to be as slow as we wish, then we can achieve C1 ερ /C(γ) ≤ C2 ερ˜, if we chose for ρ in Theorem 9 any number from the interval (˜ ρ, 1/3). The last arguments also show that the estimates ω −ω ≤ Cε and Lip (ω − ω) ≤ Cε imply that ωε − ω ≤ Cε and Lip (ωε − ω) ≤ Cρ ερ for any ρ < 1. It means that the Ampliﬁcation to Theorem 9 implies the Ampliﬁcation to Theorem 7. Finally, since linearisation of the symplectomorphism which sends solutions h(t) of (8.12) to solutions hε (t) transforms solutions of the corresponding linearised equations, then Theorem 8 follows from Theorem 10.
9
Examples
9.1
Perturbed KdV equation
Let us consider a perturbed KdV equation u˙ =
3 ∂ 1 uxxx + uux + ε fu (u, x) =: Vε (u)(x), 4 2 ∂x
(9.1)
under zero meanvalue periodic boundary conditions. In (9.1) f (u, x) is a C d smooth function (d ≥ 1), δanalytic in u. Then the nonlinear part of the vector ﬁeld Vε deﬁnes an analytic map of order one: H0d −→ H0d−1 , u −→ The equation’s hamiltonian is Hε = 17
3 ∂ uux + ε fu (u, x). 2 ∂x
2π 0
1 2 8u
! − 14 u3 − εf (u(x), x) dx.
i.e., the estimate (ωε − ω)(r1 ) − (ωε − ω)(r2 ) ≤ Cερ r1 − r2 , etc.
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For ε = 0 this is the KdV equation and deﬁned in (4.5) bounded part T 2n of any ﬁnitegap manifold TV2n satisﬁes the assumptions i)v) (see in Section 3). The linearised KdV equation has a system of Floquet solutions which is complete nonresonant (Section 6.2). The assumption 1) of Theorem 7 now holds with d1 = 3, d11 = · · · = 0 and 2) holds since dH = d˜ = 1. We get: Theorem 11. For any ρ < 1 and for suﬃciently small ε > 0, there exists a Borel subset Rεn of the cube Rn = {0 < rj < K} and a Lipschitz map Σε : Rεn × Tn −→ H0d (S 1 ), analytic in the second variable, such that: a) mesn (Rn \ Rεn ) −→ 0 as ε → 0, b) the map Σε is ερ close to the map Φ0 , Φ0 (r, z)(x) = G(Vx + z, r) (see (3.16 )), also in the Lipschitz norm, c) each torus Tεn (r) = Σε ({r} × Tn ), r ∈ Rεn , is invariant for equation (9.1) and is ﬁlled with its linearly stable timequasiperiodic solutions of the form t → Σε (r, z + tωε (r)), where the nvector ωε is Cεclose to W(r). To get the result we used Theorem 7, its Ampliﬁcation and Theorem 8. The last theorem applies since the Floquet solutions of the linearised KdV equation have real exponents and since the linearised KdV equation is well posed. The theorem implies that the union of all linearly stable timequasiperiodic solutions becomes inﬁnitedimensional and dense in H0d asymptotically as ε → 0: Corollary. The space H0d contains a subset Qε ﬁlled with linearly stable timequasiperiodic solutions of (9.1) such that its Hausdorﬀ dimension tends to inﬁnity when ε → 0 and for any ﬁxed function v ∈ H0d we have: distH0d (v, Qε ) −→ 0
as
ε → 0.
(9.2)
Proof. We deﬁne Qε as a union of all nonempty sets Σε (Rεn × Tn ) = T$ε2n , corresponding to all ngap manifolds T 2n , n = 1, 2, . . . . By Proposition 4, the set T$ε2n has positive 2ndimensional Hausdorﬀ measure when ε is small. Thus, dimH Qε −→ ∞. To prove (9.2) we note that for any µ > 0 one can ﬁnd n ≥ 1 and an ngap potential u(x) such that u − vk ≤ µ (this is a famous result of V.A.Marchenko, see [Ma], Theorem 3.4.3 and [GT], p.27). Then u equals to Φ0 (r, z) with some r ∈ R and z ∈ Tn . If ε is suﬃciently small, then by the assertion a) of the theorem, there exists r1 ∈ Rε such that r − r1  ≤ µ. Using b), we get that u − Σε (r1 , z)k ≤ Cµ + ερ and (9.2) follows since µ > 0 can be chosen arbitrary small. Another immediate consequence of the theorem is the observation that the Its–Matveev formula (4.4) with corrected frequency vector W “almost solves” the equation (9.1) for all t:
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Corollary. For any r ∈ Rn+ and any z ∈ Tn there exists an nvector Wε (r) and a solution uε (t, x) of (9.1) in H0d such that sup uε (t, ·) − 2 t
∂2 ln θ(i(V · +Wε t + z); r)d −→ 0 ∂x2
as
ε → 0.
Proof. For every ε > 0 let us choose any rε ∈ Rε such that rε → r as ε → 0 and take uε (t) = Σε (rε , z + tωε (r)). Then uε (t) − Φ0 (r, z+ωε t)k ≤ uε (t) − Φ0 (rε , z + ωε t)k + Φ0 (rε , z + ωε t) − Φ0 (r, z + ωε t)k = o(1)
as ε → 0.
2
∂ This implies the result since Φ0 (r, z + ωε t)(x) = 2 ∂x 2 ln θ(i(Vx + Wε t + z); r), where Wε = ωε .
An easy analysis of the ﬁrst step in the proof of Theorem 9 (see [KK]) shows that the new frequency vector W ε has the form W ε (r) = W (r) + εW 1 (r) + W 1!1are obtained by averaging O(ε2 ), where components W1j of the nvector 0 ! 2π − 0 f (u, x) dx . Here along the torus T n (r) 18 of the function G∗ ∂p∂ j G : (p, q, y) → u(·) is the normal form transformation from Theorem 11. Therefore the assertion of the second Corollary can be viewed as an averag∂2 ing theorem: for most r and for all z the functions 2 ∂x 2 ln θ(i(Vx+Wε t+z); r) with W ε = W (r) + εW 1 (r) + O(ε2 ) approximate solutions of the perturbed KdV equation (9.1) for all t and x, where the nvector W 1 is obtained by the averaging described above. Here “for most r” means “for all r outside a set whose measure goes to zero with ε”. 9.2
Higher KdV equations
Let us consider a perturbation of the lth equation from the KdVhierarchy: u˙ =
∂ (∇u Hl + ε∇u H1 ), ∂x
(9.3)
! 2π (l)2 u +higherorder terms with ≤ l −1 derivatives dx where Hl (u) = Kl 0 2π and H1 = 0 f (x, u, . . . , u(l−1) ) dx. The function f is assumed to be C d smooth in x, . . . u(l−1) and δanalytic in u, . . . , u(l−1) . Since ∇u H1 =
l−1 j=0
(−1)j
∂j f (j) (x, . . . , u(l−1) ), ∂xj u
then arguing as in Example 1.1, we see that order 2l − 1. 18
∂ ∂x ∇u H1
with respect to the measure (2π)−n dq = (2π)−n dz.
is an analytic map of
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Let us take a bounded part T 2n of any ngap manifold. It is invariant for the lth KdVequation (equal to (9.3)ε=0 ). As an invariant subset, it satisﬁes the assumptions i)iv). The linearised equation has a complete system of Floquet solutions (see in Section 6). Due to (4.6) this system is nonresonant. Now Theorem 7 applies to equation (9.3) since the assumption 1) holds with d1 = 2l + 1, d11 = · · · = 0 (see (6.20)) and 2) holds with dH = d˜ = 2l + 1. We see that most of ngap solutions of the lth KdV equation persist in the perturbed equation (9.3) with suﬃciently small ε in the same sense as for the KdV equation. 9.3
Perturbed SG equation
Since the system of ﬁnitegap solutions (4.7), (4.8) of the SG equation under the even periodic boundary conditions satisfy the assumptions i)iv) (Section 4) and the system of Floquet solutions (6.23) is complete nonresonant, then Theorem 7 applies to the perturbed equation,
= εf (u, x), utt − uxx + sin u (9.4) u(t, x) ≡ u(t, x + 2π) ≡ u0 (−x), where f is C k smooth function (k ≥ 1), analytic in u. Accordingly, for any n ≥ 1, most of ﬁnitegap solutions (4.7), (4.8) persist as timequasiperiodic solutions of (9.4). The persisted solutions are Lyapunovstable if the open gaps for solutions (4.7) are suﬃciently small. Otherwise, they in general are nonstable. Since it is unknown if the ﬁnitegap solutions of (SG) jointly are dence in a function space, then we do not know if the persisted solutions of (9.4) are asymptotically dense as ε → 0 (cf. (9.2)). 9.4
KAMpersistence of lowerdimensional invariant tori of nonlinear ﬁnitedimensional systems
Let R2N" be an euclidean space, given the usual symplectic structure, let T 2N = r∈R Trn be an analytic submanifold of R2N , diﬀeomorphic to R×Tn , R Rn , and H1 , . . . , Hn be commuting hamiltonians, as in Proposition 3 (so they are deﬁned and analytic in the vicinity of T 2n and each torus Trn is invariant for every hamiltonian vector ﬁeld VHj ). Let us take any hamiltonian – say, H1 . Then the vector ﬁeld VH1 T 2n has the form ωl (r)∂/∂zl and by Proposition 3 linearised equations have Floquet solutions with some frequencies νj (r), which are real analytic functions. Applying Theorem 7 we get that: Theorem 12. Let us assume that the following analytic functions do not vanish identically: l · ν(r) + s · ω(r),
l ∈ ZN −n , 1 ≤ l ≤ 2; s ∈ Zn .
(9.5)
KAMpersistence of ﬁnitegap solutions
121
Let h be an analytic function, deﬁned in the vicinity of T 2n . Then most of the tori Trn persist as invariant ntori of the perturbed Hamiltonian vector ﬁeld VH1 +εh , 0 < ε ( 1, in the sense, speciﬁed in Theorem 7. The persisted tori are ﬁlled with quasiperiodic solutions with zero Lyapunov exponents. This reduction of the Main Theorem is much easier than the Main Theorem itself. Its claim remains essentially true under weaker assumptions: it is suﬃcient to check that only functions (9.5) with l = 1 do not vanish identically, see [Bour] (we note that under this weaker assumption the claim about Lyapunov exponents is not true).
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[DMN] Dubrovin B.A., Matveev V.F., Novikov S.P., Nonlinear equations of Kortewegde Vries type, ﬁnite zone linear operators, and Abelian varieties. Uspekhi Mat. Nauk 31:1 (1976) 55–136; English transl. in Russ. Math. Surv. 31:1 (1976) [EFM] Ercolani, N., Forest, M.G., McLaughlin, D., Geometry of the modulational instability. Part 1. Preprint, University of Arizona, 1987 [Fal] Falconer K., Fractal Geometry. l John Wiley & Sons, 1990 [Go] Godbillon C., G´eom´ etrie diﬀ´erentielle et m`ecanique analytique. Hermann, Paris, 1969 [GR] Gunning R.C., Rossi H., Analytic functions of several complex variables. PrenticeHall, Inc., Englewood Cliﬀs, N.J., 1965 [GS] Guillemin V., Sternberg H., Geometric asymptotics. Amer. Math. Soc., Providence, 1977 [GT] Garnet J., Trubowitz, E., Gaps and bands on one dimensional Schr¨ odinger operator II. Comment. Math. Helvetici 62 (1987) 18–37 [HS] Halmos P., Sunder V. Bounded integral operators on L2 spaces. SpringerVerlag, Berlin, 1978 [K] Kuksin S.B., Nearly integrable inﬁnitedimensional Hamiltonian systems. Lecture Notes Math. 1556. SpringerVerlag, Berlin, 1993 [KK] Kuksin S.B., Analysis of hamiltonian PDEs. Manuscript [K1] Kuksin S.B., Perturbation theory for quasiperiodic solutions of inﬁnitedimensional Hamiltonian systems, and its applications to the Kortewegde Vries equation. Matem. Sbornik 136(178):3 (1988); English transl in Math. USSR Sbornik 64 (1989) 397–413 [K2] Kuksin S.B., A KAMtheorem for equations of the Korteweg–de Vries type. Rev. Math. & Math. Phys. 10:3 (1998) 1–64 [K3] Kuksin S.B., An inﬁnitesimal Liouville–Arnold theorem as criterion of reducibility for variational Hamiltonian equations. Chaos, Solitons and Fractals 2:2 (1992) 259269 [Kap] Kappeler T., Fibration of the phase space for the Korteweg–de Vries equation. Ann. Inst. Fourier 41 (1991) 539–575 [Kat1] Kato T., Quasilinear equations of evolutions, with applications to partial diﬀerential equations. Lecture Notes Math. 448, 25–70, SpringerVerlag, Berlin, 1975 [Kat2] Kato T. Perturbation theory for linear operators. SpringerVerlag, Berlin, 1966 [LM] Lions J.L., Magenes E., Probl´emes aux limites non homog´ enes et applications. Dunod, Paris, 1968; English transl. SpringerVerlag, Berlin, 1972 [Ma] Marˇcenko, V.A., SturmLiouville operators and applications. Naukova Dumka, Kiev, 1977 Russian; English transl. Birkh¨ auser, Basel, 1986 [McK] McKean H.P., The sineGordon and sinhGordon equation on the circle. Comm. Pure Appl. Math. 34 (1981) 197–257 [Mil] Milnor J., Singular points of complex hypersurfaces. University Press, Princeton, 1968 [Mo] Moser J., Various aspects of integrable hamiltonian systems. In Proc. CIME Conference, Bressansone, Italien, 1978. Progr. Math. 8. Birkh¨ auser, 1980 [MT] McKean H.P., Trubowitz E., Hill’s operator and hyperelliptic function theory in the presence of inﬁnitely many branch points. Comm. Pure Appl. Math. 29 (1976) 143–226
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Novikov S.P., A periodic problem for the Kortewegde Vries equation, I. Funktsional. Anal. i Prilozhen. 8:3 (1974) 54–66; English transl in Functional Anal. Appl. 8, 236–246 (1974) Nekhoroshev N.N., The Poincar´eLyapunovLiouvilleArnold theorem, Funktsional. Analiz i Priloz. 28:2 (1994); English transl in Functional Anal. Appl. 28, 128–129 (1994) P¨ oschel J., A KAMtheorem for some nonlinear PDEs. Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15 Pazy A., Semigroups of linear operators and applications to partial diﬀerential equations. SpringerVerlag, Berlin, 1983 P¨ oschel J., Trubowitz E., Inverse spectral theory. Academic Press, New York–London, 1987 Reed M., Simon B., Methods of modern mathematical physics. Vol. 2. Academic Press, New York–London, 1975 Reed M., Simon B., Methods of modern mathematical physics, Vol. 4, Academic Press, New York–London, 1978 Springer G., Introduction to Riemann surfaces, second ed.. Chelsea Publ. Company, New York, 1981 Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskij L.P., Theory of solitons. Nauka, Moscow, 1980, English transl. Plenum Press, New York, 1984
Analytic linearization of circle diﬀeomorphisms JeanChristophe Yoccoz
1
Introduction
Let T = R/Z and r ∈ {0, +∞, ω} ∪ [1, +∞). We denote Diﬀr+ (T) the group of homeomorphisms of T of class C r and C r isotopic to the identity (if r = 0 it is the group of homeomorphisms of T; if r ≥ 1, r ∈ R∗ \ N, it is the group of C r diﬀeomorphisms whose rth derivative veriﬁes a H¨older condition of exponent r − )r%; if r = ω it is the group of Ranalytic diﬀeomorphisms). We denote Dr (T) the group of C r diﬀeomorphisms f of the real line such that f − id is Zperiodic. One can embed R into Dω (T) as the subgroup of translations α ∈ R → Rα ∈ Dω (T)
where
Rα : x → x + α,
and one can embed the quotient T = R/Z into Diﬀω + (T) as the subgroup of rotations α ∈ T → Rα ∈ Diﬀω + (T)
where
Rα : x → x + α(mod1).
One has Diﬀr+ (T) " Dr (T)/{Rp , p ∈ Z}. We consider the usual C r topology on Dr (T) and Diﬀr+ (T). Rotations on T have very simple dynamics. Poincar´e asked under which condition a given homeomorphism f of T is equivalent (in some sense, e.g.: measurably, topologically, smoothly, analytically, etc.) to some rotation. He also gave a ﬁrst answer to this question, today known as Poincar´e’s classiﬁcation theorem: if the rotation number of ρ (f ) is irrational then f is combinatorially conjugated to the rotation Rρ(f ) (see Corollary 3.3). In Section 3.3 we explain the Denjoy theory [De]. Fixing the rotation number to be irrational Denjoy proved that if we add regularity to the homeomorphism f (actually f is C 1 and Df has bounded variation) then f is topologically conjugated to the rotation Rρ(f ) . This theorem doesn’t give us information about the regularity of the conjugacy h. Denjoy proved also that the hypothesis f ∈ C 1+BV cannot be relaxed too much, constructing examples of C 1 diﬀeomorphisms which are not topologically conjugated to a rotation (see Section 3.4). In [He1] there is a generalisation of this result: for each irrational α there exists a dense subset, in
L.H. Eliasson et al.: LNM 1784, S. Marmi and J.C. Yoccoz (Eds.), pp. 125–173, 2002. c SpringerVerlag Berlin Heidelberg 2002
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the space of C 1 diﬀeomorphisms whose rotation number is α, such that every element of this subset is not topologically conjugated to a rotation. The step to higher order diﬀerentiability for the conjugation h requires new techniques and additional hypotheses on the rotation number. The fundamental regularity criterion is the following: Let k ≥ r ≥ 1, f ∈ Diﬀ k+ (T). f is C r conjugated to a rotation (then to the rotation Rρ(f ) ) if and only if the family (f n )n≥0 is bounded in the C r topology. In [Ar1] it was proved that if the rotation number veriﬁes a diophantine condition and if the analytic diﬀeomorphism f is close enough to a rotation, then the conjugation is analytic. At the same time examples of analytic diffeomorphisms, with irrational rotation number, for which the conjugation is not even absolutely continuous were given. In [He1] Arnol’d’s result was improved obtaining a global result: there exists a set A ⊂ (0, 1), with full Lebesgue measure, such that if the rotation number of the C ∞ diﬀeomorphism belongs to A, then it is C ∞ conjugated to a rotation. A similar result holds for ﬁnitely diﬀerentiable diﬀeomorphisms, but in this case the conjugacy is less regular: this phenomenon of loss of diﬀerentiability is typical of small divisors problems. In the case of ﬁnite diﬀerentiability, the following result is due to Y. Katznelson and D. Ornstein [KO1,KO2], see also [Yo1,KS,SK]. Theorem 1.1 Let f be a C k circle diﬀeomorphism, k ∈ R, k > 2, with rotation number α. Assume that α veriﬁes the following diophantine condition: there exists c > 0 and τ ≥ 0 such that for all n ∈ N, n > 0 e2πinα − 1 > cn−τ −1 . Then the homeomorphism h which conjugates f to Rα is of class C k−1−τ − for all > 0. If f is C 2 and α is a number of constant type (i.e. τ = 0) then h is absolutely continuous. The previous result shows that the loss of diﬀerentiability is at most 1 + τ + . In Section 4 we will present our results in the analytic case. We distinguish between local and global situation. In the local case one must assume that the rotation number α belongs to the set B of Brjuno numbers (see Section 2.4). We have two statements: in the ﬁrst one we assume that the diﬀeomorphism is analytic and univalent on some “big” complex strip B∆ = {z ∈ C : m z < ∆}, in the second one that the diﬀeomorphism is analytic in some “thin” complex strip Bδ but it is very close to a rotation: Theorem 1.2 Let f ∈ Diﬀ ω + (T), ρ (f ) = α ∈ B and assume that ∆ > 1 B (α) +c, where c is a universal constant. If f is analytic and univalent 2π in B∆ then there exists h ∈ Diﬀ ω + (T), analytic in B∆ , with ∆ = ∆ − 1 B (α) −c, which conjugates f to R in B . α ∆ 2π
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Theorem 1.3 Let f ∈ Diﬀ ω + (T), ρ (f ) = α ∈ B. Given δ > 0 there exists 0 (α, δ) such that if f is holomorphic in Bδ and there f (z) − z − α ≤ 0 −1 then there exists h ∈ Diﬀ ω on B δ . + (T) such that f = h ◦ Rα ◦ h 2
Using the theorem (proven in [Yo2]) showing that the Brjuno condition is also necessary for the problem of linearization of germs of complex analytic diﬀeomorphisms of (C, 0), and following the construction described in [PM], which associates to each nonlinearizable germ of diﬀeomorphism of (C, 0) a nonlinearizable analytic diﬀeomorphism of T close to a rotation, one sees that the Brjuno condition is necessary also in our context. In the global case we introduce a condition (condition H, see Section 2.5) which is suﬃcient and in some sense also necessary, more exactely: Theorem 1.4 Let f ∈ Diﬀ ω + (T) and let α be its rotation number. If α ∈ H, there exists h ∈ Diﬀ ω (T) such that f = h◦Rα ◦h−1 . Moreover if α ∈ H there + ω exists f ∈ Diﬀ + (T) with rotation number α, which is not C ω linearizable. Theorems 1.2 and 1.3 will be proved in Sections 4.2 and 4.3 respectively. The ﬁrst part of Theorem 1.4 will be proved in Section 4.5, using an extension of Denjoy’s theory described in Section 4.4. The last Section 4.6 is devoted to the construction of counterexamples showing that the assumption α ∈ H in the global theorem is also necessary.
2 2.1
Arithmetics Introduction
The aim of this section is to introduce some elementary but fundamental facts from the theory of approximation of irrational numbers by rationals. The most important tool we will use here is the continued fraction expansion of a real number (see Section 2.2 and [HW]). As it will be clear in what follows, the action of the modular group GL (2, Z) on R = R ∪ {∞} plays a fundamental role in the renormalization procedure we will use to study the linearization problem. To better understand this action one can introduce a fundamental domain [0, 1) for one of the two generators (the translation T ) and concentrate the attention to the inversion x → 1/x restricted to [0, 1). Then one translates back 1/x to the fundamental domain and iterates the process. This gives an expanding map on [0, 1), the Gauss’ map A (2.1). As A is more and more expanding as x → 0+ one obtains a “microscope” and the symbolic dynamics of this map gives rise to the continued fraction algorithm. The convergents of the continued fraction expansion of an irrational number give its best rational approximations in a very precise sense (see Lemma 2.1). The classical diophantine conditions are reviewed in Section 2.3. They can be recasted in terms of the speed of growth of various quantities related to the continued fraction expansion. This opens the way to introducing two sets of
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“badly approximable” irrational numbers which include diophantine numbers and which we will prove being the optimal sets for the linearization problem (respectively local for the set B, introduced in Section 2.4, and global for the set H, introduced in Section 2.5). 2.2
Continued Fractions
We consider the map: A : (0, 1) → (0, 1) deﬁned by: A (x) =
1 213 − x x
(2.1)
To each α ∈ R \ Q, iterating A, we associate its continued fraction expansion as follows. Let (αn )n≥0 be deﬁned by: α0 = α − )α%,
αn = An (α0 )
for n > 0
(2.2)
∀n ≥ 1
(2.3)
and (an )n≥0 by a0 = )α%
−1 αn−1 = an + αn
then 1
α = a0 + a1 +
1 ..
.+
1 an + αn
or shortly α = [a0 , a1 , . . . , an + αn ]. The nth convergent is deﬁned by pn = [a0 , a1 , . . . , an ] = a0 + qn
1 a1 +
(2.4)
1 ..
.+
1 an
It is easy to identify the numerator and denominator of the nth convergent with the sequences (pn )n≥−2 and (qn )n≥−2 , deﬁned by: p−2 = 0, p−1 = 1, and q−2 = 1, q−1 = 0, and Moreover α =
pn +pn−1 αn qn +qn−1 αn
pn = an pn−1 + pn−2 ,
for all n ≥ 0
qn = an qn−1 + qn−2 ,
for all n ≥ 0
qn α−pn and αn = − qn−1 α−pn−1 .
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Let β−1 = 1
βn =
n
n
αj ≡ (−1) (qn α − pn )
∀n ≥ 0
(2.5)
j=0
then qn+1 βn + qn βn+1 = 1 an+1 βn + βn+1 = βn−1 and as an ≥ 1 for all n −1
(qn + qn+1 )
−1
< βn < (qn+1 )
.
(2.6)
We then have the following standard results: Lemma 2.1 (Best approximation) Let q ≥ 0, p ∈ Z, n ≥ 0; if qα − p is strictly between qn α − pn and qn+1 α − pn+1 then either q ≥ qn + qn+1 or p = q = 0. We refer to [HW] for the elementary proof. Corollary 2.2 Consider the set {qα − p; 0 ≤ q < qn+1 , p ∈ Z}, ordered as . . . < x−1 < 0 = x0 < x1 < . . . . Let xk = qα − p be an element of this set. Then
xk+1 = xk + βn if q < qn+1 − qn • if n is even xk+1 = xk + βn + βn+1 if q ≥ qn+1 − qn
xk−1 = xk + βn if q < qn+1 − qn • if n is odd xk−1 = xk + βn + βn+1 if q ≥ qn+1 − qn Proof. We consider only the case n is even, the odd case being similar. We write xk+1 = q α − p , 0 ≤ q < qn+1 . If q < qn+1 − qn , xk + βn = (q + qn )α − (p + pn ) belongs to the set. If we had xk+1 < xk + βn , both q ≥ q (xk+1 being between xk and xk + βn ) and q ≤ q (xk + βn being between xk+1 and xk+1 + βn ) would be impossible by Lemma 2.1, a contradiction. If q ≥ qn+1 − qn , xk + βn + βn+1 belongs to the set. If we had xk+1 < xk + βn + βn+1 , as q ≥ 0 > q − qn+1 , xk+1 would be between xk = (xk + βn+1 ) − βn+1 and (xk + βn+1 ) + βn and we would have by Lemma 2.1 q ≥ q + qn ≥ qn+1 , again a contradiction. Let us consider the standard action of P GL (2, Z) = GL (2, Z) /{±I} on R = R ∪ {∞} given by: a b αa + b for g = ∈ GL (2, Z) , g · α = . (2.7) αc + d c d Then we have
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Proposition 2.3 Two irrational numbers α and α belong to the same GL (2, Z)orbit if and only if there exists n ≥ 0 and n ≥ 0 such that αn = αn . 2.3
Diophantine Conditions
Given τ ≥ 0, α ∈ R \ Q satisﬁes a diophantine condition of order τ (for short α ∈ CD (τ )) if there exists a constant c > 0 such that: % c p p %% % (2.8) %α − % ≥ 2+τ ∀ ∈ Q, q ≥ 1. q q q Proposition 2.4 Let τ ≥ 0 and α ∈ R \ Q then the following statements are equivalent 1. 2. 3. 4. 5.
α ∈ CD (τ); qn+1 = O qnτ +1 ; an+1 = O(qnτ ); −1 αn+1 = O(βn−τ ); −1 βn+1 = O βn−τ −1 ;
where (αn )n≥0 , (pn /qn )n≥0 , (βn )n≥−1 and (an )n≥1 are deﬁned in Section 2.2. The proof is elementary and is left as an exercise. We say that α satisﬁes a diophantine condition if α ∈ CD = ∪τ ≥0 CD (τ ). For all τ , CD (τ ) is invariant for the action of GL (2, Z). For all τ > 0, CD (τ ) has full measure; on the other hand (R \ Q) \ CD is a Gδ dense subset of R and CD (0) has zero measure. 2.4
Brjuno function and condition B
Let α ∈ R \ Q, we deﬁne (following [Yo2] and [MMY1]) the Brjuno function B : R \ Q → R+ ∪ {+∞} B (α) = βn−1 log αn−1 (2.9) n≥0
where (αn )n≥0 and (βn )n≥−1 are deﬁned in Section 2.2. We say that α ∈ R \ Q satisﬁes the condition B (or is a Brjuno number, for short α ∈ B) if B (α) < +∞. Remark 2.5 As in [Yo2] and [MMY1] we extend the above deﬁnition to rational values setting B (α) = +∞ or e− B(α) = 0 for α ∈ Q. The following proposition collects some properties of the Brjuno function (see [MMY1], Proposition 2.3, p. 277, for a proof). Proposition 2.6 The function B satisﬁes
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1. B (α) = B (α + 1) for all α ∈ R; 2. for all α ∈ (0, 1) then B (α) = log α−1 + α B α−1 ; 3. there exists a constant C1 > 0 independent of α such that for all α ∈ R \ Q % log qj+1 %% % (2.10) % ≤ C1 % B (α) − qj j≥0
where (qj )j≥0 are the denominators of the convergent of α deﬁned in (2.4). Remark 2.7 By 2. one has that B is P GL (2, Z)invariant. Indeed the Brjuno function B is a P GL (2, Z)cocycle (see [MMY2], Appendix 5). Remark 2.8 Let α ∈ CD, by Proposition 2.4 there exists τ ≥ 0 such that qn+1 = O qnτ +1 , thus logqqnn+1 ≤ (1 + τ ) logqnqn + qcn and then B (α) < +∞. Therefore CD ⊂ B and the set B has full measure. Remark 2.9 For positive τ we could deﬁne a condition CD (τ ) using the − τ1 function B (τ ) (α) = (note that the function B (τ ) (α) = j≥0 βj−1 αj −1 τ would have the same features) as follows: j≥0 βj−1 αj α ∈ CD (τ ) if and only if B (τ ) (α) < +∞. Then the condition CD (τ ) is almost equivalent to condition CD (τ ), in fact CD (τ ) ⇒ CD (τ ) , CD (τ ) ⇒ CD (τ ) 2.5
∀τ > τ.
Condition H
For α ∈ (0, 1), x ∈ R, deﬁne
α−1 (x − log α−1 + 1) if x ≥ log α−1 , rα (x) = ex if x ≤ log α−1 .
(2.11)
Observe that rα is of class C 1 on R, satisfying rα (log α−1 ) = Drα (log α−1 ) = α−1 ; ex ≥ rα (x) ≥ x + 1 for all x ∈ R; Drα (x) ≥ 1 for all x ≥ 0. For α ∈ R \ Q, we now set, for k > 0 ∆k (α) = rαk−1 ◦ · · · ◦ rα0 (0). An easy calculation gives
(2.12) (2.13) (2.14)
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Lemma 2.10 Let α ∈ B and k ≥ 0. Assume that B(αk ) ≤ ∆k (α). Then one has, for l ≥ k βl−1 ∆l (α) = βk−1 ∆k (α) +
l−1
βj−1 −
j=k
= βk−1 (∆k (α) − B(αk )) +
l−1
βj−1 log αj−1
j=k l−1
βj−1 + βl−1 B(αl )
j=k
≥ βl−1 B(αl ) ≥ βl−1 log αl−1 . Let us deﬁne, for n ≥ k ≥ 0: Hk,n = {α ∈ B, B(αn ) ≤ ∆k (αn−k )}, and then H = ∩m≥0 (∪k≥0 Hk,k+m ) = {α ∈ B, ∀m ≥ 0, ∃k ≥ 0, B(αm+k ) ≤ ∆k (αm )} Proposition 2.11 1. For 0 < k ≤ n, one has Hk−1,n ⊂ Hk,n ⊂ Hk+1,n+1 ; 2. Let α ∈ R \ Q. Then α ∈ H (resp. α ∈ B) if and only if the orbit of α under GL (2, Z) is contained in (resp. intersects) ∪k≥0 Hk,k . 3. The subset H of R is a Fσδ subset (i.e. a countable intersection of Fσ subsets). Proof. 1. From Lemma 2.10 we have Hk,n ⊂ Hk+1,n+1 . Also ∆1 (αn−k ) = 1 and from (2.14) it follows that ∆k (αn−k ) ≥ ∆k−1 (αn−k+1 ) + 1 and thus Hk−1,n ⊂ Hk,n . 2. Let α ∈ R \ Q. Then α ∈ H if and only if αn ∈ ∪k≥0 Hk,k for all n ≥ 0. As the αn ’s belong to the orbit of α under GL (2, Z), we see that if the orbit of α is contained in ∪k≥0 Hk,k then α ∈ H. Conversely, if α ∈ H and α belongs to the orbit of α, by Proposition 2.3 there exist integers n and n such that αn = αn . Then αn ∈ ∪k≥0 Hk,k and it follows from 1. that also α ∈ ∪k≥0 Hk,k . On the other hand, we have by deﬁnition ∪k≥0 Hk,k ⊂ B, and B is invariant under GL (2, Z). Conversely, if α ∈ B, let k ≥ B(α) and α ∈ R \ Q be such that αk = α0 . Then α belongs to the orbit of α and we have by (2.13) ∆k (α ) ≥ k ≥ B(αk ) hence α ∈ Hk,k . 3. As B is lower semicontinuous, for every n ≥ k ≥ 0, Hk,n is a closed subset and the conclusion follows. Remark 2.12 One has the obvious inclusions CD ⊂ H ⊂ B. Moreover, by 2. H is GL (2, Z)invariant. Example 2.13 Consider the compact subset K ⊂ [0, 1] whose elements are irrationals α such that a1 = 1,
ai+1 ≤ exp (2ai ) − 1.
Analytic linearization of circle diﬀeomorphisms
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Let α ∈ K, j ≥ 0; denote by p˜n /˜ qn the convergents of αj . For n > 0, we have −1 −1 αj+l log αj+n ≤ q˜n−1 log(aj+n+1 + 1) ≤ 2˜ qn−1 aj+n ≤ 2˜ qn−1 ; 0≤l log α−1 ; as x < rα (x) < ex , the number t = L(rα (x)) − L(x) belongs to (0, 1). Consider the map w → Et (w) − rα (w) for w ≥ log α−1 . It is convex, negative at log α−1 , and vanishes at x; therefore the derivative at x is positive, and thus we must have D(L ◦ rα − L)(x) < 0. Lemma 2.15 Let α ∈ B. 1. For any k ≥ 0, we have B(αk+1 ) ≤ exp(B(αk ) − 1). 2. If α ∈ / ∪k≥0 Hk,k , the positive sequence L(B(αk )) − L(∆k (αk )) is decreasing. Proof. 1. The assertion follows immediately from the relation B(αk+1 ) = eu (B(αk ) − u),
u = log αk−1
maximizing over u. 2. From the ﬁrst assertion we have always B(αk+1 ) ≤ rαk (B(αk ) − 1) ≤ rαk (B(αk )). Therefore the second assertion of Lemma 2.15 follows from the second assertion of Lemma 2.14. We now consider an irrational number α ∈ B. Clearly, if α ∈ / H, we must have lim B(αk ) = +∞.
k→∞
Fix c0 > 0 large enough such that, for any irrational α, ˜ 1/2 ≤ c0 . β˜ k−1
k≥0
We deﬁne, starting with k0 = 0, an increasing sequence of indices ki setting 1/2 βki ki+1 = inf{k > ki , log αk−1 ≥ c−1 B(αki +1 )}. 0 βk−1
Analytic linearization of circle diﬀeomorphisms
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The set on the righthand side is indeed nonempty: otherwise, as B(αki +1 ) =
βk−1 log αk−1 βki
k>ki
we would have
B(αki +1 ) < c−1 0
βk−1 1/2 B(αki +1 ) ≤ B(αki +1 ). βki
k>ki
Lemma 2.16 For α ∈ B and ki as above we have ) = 0. lim L(B(αki )) − L(log αk−1 i
i→+∞
If α ∈ /H lim L(B(αki +1 )) − L(B(αki )) = 1.
i→+∞
Proof. For the ﬁrst limit, we note that βki βki+1 −1
B(αki +1 ) ≥ B(αki+1 ) ≥
log αk−1 i+1
≥
c−1 0
βki
1/2 B(αki +1 )
βki+1 −1
(2.15) But it is easily checked that lim
x≥1,y≥1,xy→∞
1/2 L(xy) − L(c−1 y) = 0 0 x
from which the assertion follows. The second part of the Lemma is then an immmediate consequence of the second part of Lemma 2.15: as α ∈ / H the sequence L(B(αk )) − L(∆k (α)) is positive decreasing; setting εi = L(B(αki )) − L(log αk−1 ), we have that, if i L(∆ki (α)) ≤ L(B(αki )) − εi , then L(∆ki +1 (α)) = L(∆ki (α)) + 1 ≤ L(B(αki +1 )). This forces the second assertion of the Lemma.
We now consider the two relations (2.15) and βki 1 1 1 = βki+1 −1 βki+1 −1 βki −1 αki
.
We get 4 lim
L(βk−1 ) i+1 −1
− max
L(βk−1 ), L(αk−1 ), L i −1 i
β ki βki+1 −1
5 =0
136
and
J.C. Yoccoz
lim L(B(αki+1 )) − max L 4
βki βki+1 −1
5 , L(B(αki +1 ))
=0
which, with Lemma 2.16, transform to 4 5 β ki −1 −1 −1 0 = lim L(βki+1 −1 ) − max L(βki −1 ), L(log αki ) + 1, L βki+1 −1 4 5 βk i −1 −1 = lim L(log αki+1 ) − max L , L(log αki ) + 1 . βki+1 −1 We claim that Proposition 2.17 We have limi→+∞ L(log αk−1 ) − L(βk−1 ) = 0. i+1 i+1 −1 Proof. If not we would have ) > L(log αk−1 )+1 L(βk−1 i −1 i for inﬁnitely many i’s. But setting β ki −1 −1 ) − max L(β ), L(log α ) + 1, L εi = L(βk−1 ki −1 ki i+1 −1 βki+1 −1 βk i −1 εi = L(log αk−1 , L(log α ) − max L ) + 1 ki i+1 βki+1 −1 ηi = L(βk−1 ) − L(log αk−1 ) i −1 i we have that if ηi+1 > εi − εi then −1 −1 max L(βki −1 ), L(log αki ) + 1, L and L(βk−1 ) i −1
ηi+1 = εi −
εi
+
≤ εi −
εi
+ ηi − 1.
− max L
β ki βki+1 −1 βk i
βki+1 −1
= L(βk−1 ) i −1
, L(log αk−1 ) i
+1
Corollary 2.18 Let α ∈ / H. Then for any t < 1 there are inﬁnitely many rationals such that α − p/q ≤ (Et (q))−1 . 2.6
Z2 actions by translations and continued fractions
Let (e−1 , e0 ) be the canonical basis of Z2 . We consider the following action π Z2 → Homeo+ (R): to (e−1 , e0 ) we associate two commuting homeomorphisms by
π(e−1 ) : x → x − 1 = T (x) (2.16) π(e0 ) : x → x + α = f (x)
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We perform the sequence of changes of basis of Z2 given by
en−1
en
e−1
= Mn
,
e0
where
Mn =
pn−1 qn−1 pn
,
qn
thus
en
en+1
= An+1
en−1 en
,
where
An+1 =
0
1
1 an+1
.
The homeomorphisms corresponding to the new basis (en−1 , en ) will be (π(en−1 ), π(en )) where π(en ) :
x → x + (−1)n βn .
We can now rewrite Lemma 2.1 (from now on we will use the multiplicative notation for Zk actions): Lemma 2.19 (Best approximation 2nd version) Let p, q ∈ Z, if qα − p is strictly between en · 0 and en−1 · 0 then either q ≥ qn + qn−1 or p = q = 0.
3 3.1
The C r theory for r ≥ 0 The C 0 theory
We consider an action of Zk on R, i.e. an homomorphism π : Zk → Homeo+ (R). We will assume that the action has no common ﬁxed point. We deﬁne Z + = {e ∈ Zk : e · x > x ∀x ∈ R}; we have then −Z + = {e ∈ Zk : e · x < x ∀x ∈ R}.
Proposition 3.1 The subset Z + of Zk has the following properties: 1. 2. 3. 4. 5.
Z + = ∅; −Z + ∩ Z + = ∅; if e ∈ −Z + and e ∈ Z + then e + e ∈ Z + ; for any r ≥ 1, e ∈ Z + if and only if re ∈ Z + ; for any e ∈ Z + , e ∈ Zk there exists n such that e + ne ∈ Z + .
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Proof. 1. We prove by induction on k that if Z + = ∅ the action has a common ﬁxed point. Write e1 , . . . , ek for the canonical basis of Zk . When k = 1, e1 must have a ﬁxed point if neither e1 nor −e1 belong to Z + . From the induction hypothesis, we know that the closed set F of points ﬁxed by e1 , . . . , ek−1 is non empty; also ek has a ﬁxed point x0 . If x0 ∈ F , we are done; otherwise, as ek commutes with ei , i < k, the set F is ek invariant, and the component of R \ F containing x0 is ﬁxed by ek . But then an endpoint of this component in F is ﬁxed by ek . 2. Obvious 3. By assumption, there exists x0 such that e · x0 ≥ x0 . As e ∈ Z + , and thus has no ﬁxed point, there exists for any x ∈ R an integer n ∈ Z such that ne · x0 ≤ x < (n + 1)e · x0 . Then (e + e ) · x ≥ [(n + 1)e + e ] · x0 ≥ (n + 1)e · x0 > x. 4. Obvious 5. Let x0 ∈ R; as above there exists n ∈ Z such that (n − 1)e · (e · x0 ) ≥ x0 . Then (n − 1)e + e ∈ / −Z + and ne + e ∈ Z + by 3. Remark 3.2 A subset Z + ⊂ Zk satisﬁes the properties 1–5 of Proposition 3.1 if and only if there exists a non zero linear form ρ : Zk → R such that Z + = {e ∈ Zk , ρ(e) > 0}, the form ρ being then uniquely determined up to a positive scalar multiple. We leave the proof as an exercise. The form ρ may be deﬁned as follows. Choose e0 ∈ Z + and deﬁne, for e ∈ Zk N + (e) = inf{n ∈ Z, ne0 − e ∈ Z + } N − (e) = sup{n ∈ Z, ne0 − e ∈ −Z + } Then N + (ke) N − (ke) = lim . k→∞ k→∞ k k
ρ (e) = lim
(3.1)
Fix such a linear form ρ and consider the action by translations πρ (e) · x = x + ρ(e),
for e ∈ Zk , x ∈ R.
Given x0 ∈ R, ρ vanishes on the πstabilizer of x0 ; this allows to deﬁne a map h from the πorbit of x0 onto ρ(Zk ) by h(e · x0 ) = ρ(e). This map is immediately seen to be order preserving. When the Zrank of ρ is > 1, ρ(Zk ) is dense in R, and h has a unique order preserving extension to R, which is automatically continuous and surjective and still a semiconjugacy from π to πρ . We thus have
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Corollary 3.3 (Poincar´e) When the Zrank of ρ is > 1, there is an order preserving surjective continuous semiconjugacy h from π to πρ : h(e · x) = h(x) + ρ(e), ∀x ∈ R, ∀e ∈ Zk uniquely determined up to an additive constant. 3.2
Equicontinuity and topological conjugacy
Let F be an element of Diﬀr+ ( T) and f ∈ Dr (T) a lift of F . To such an F we can associate the Z2 action π deﬁned as follows: to the canonical basis of Z2 we associate
f−1 : x → x − 1 = T (x) , (3.2) : x → f (x) . f0 Then ρ (π) = (−1, α) and the number α ∈ R is called the rotation number of f (or associated to the Z2 action π; we will also write α = ρ(f ) and we deﬁne the rotation number of F being ρ(F ) = α(mod1)). Poincar´e’s result (Corollary 3.3) can be recasted in the following form: Proposition 3.4 (Poincar´e) Let f ∈ D0 (T), α = ρ(f ) ∈ R \ Q. There exists h : R → R continuous, monotone nondecreasing such that h−id is Zperiodic and h ◦ f = Rα ◦ h. Remark 3.5 By Proposition 3.4 µ = Dh is an F invariant probability measure on T and h is a homeomorphism if and only if the support of µ is T1 . Let s ∈ {0, +∞, ω} ∪ [1, +∞), s ≤ r. The fundamental criterion of C s conjugacy of a diﬀeomorphism F to the rotation Rα , where α = ρ(F ), has been stated in the Introduction. Here, following [He1] (Section II.9) we will prove a criterion that guarantees that F is topologically conjugated to Rα . For the general case we refer to [He1] (Chapitre IV). Proposition 3.6 Let F ∈ Diﬀr+ (T), α = ρ(F ). F is C 0 conjugated to Rα if and only if the family (F ni )i∈N is equicontinuous for some ni → ∞. Proof. The condition is clearly necessary: here we assume that α ∈ R \ Q and we prove that it is also suﬃcient. We leave to the reader the case α ∈ Q. Let µ be an F invariant probability measure. By Remark 3.5, F is topologically conjugated to Rα if and only if suppµ = T1 . If one has K = suppµ = T1 , since K is F invariant, T1 \ K is open and F invariant. Let J be a connected component of T1 \ K. Then the intervals (F n (J))n∈Z are pairwise disjoint connected components of T1 \ K. Thus F n (J) → 0 as n → ±∞, but this contradicts the equicontinuity of (F n )n∈N .
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The Denjoy theory.
Let F be an element of Diﬀ 2+ (T) and f a lift of F . Set k = 2 and consider π We assume that the rotation the action Z2 → Diﬀ 2+ (R) deﬁned as in (3.2). ! pn will be the convergents of α number α ∈ R \ Q ∩ (0, 1). As usual qn n≥0
and we can deﬁne a new basis of Z2 by (fn−1 , fn ) where fn = f qn T pn . The goal of this section is to prove Denjoy’s Theorem (Corollary 3.13): if the action is by C 2 diﬀeomorphisms then the semiconjugacy h of Corollary 3.3 is indeed a conjugacy (there is no interval on which h is constant). The fundamental step will be Corollary 3.12 which allows to obtain a bound for Dfn C 0 . The following lemma is fundamental for all the theory, in fact it gives control of the reciprocal position of the iterates of a ﬁxed interval. Lemma 3.7 Let In (x) = [x, fn (x)] and Jn (x) = In (x) ∪ In fn−1 (x) = [fn−1 (x), fn (x)]. Then 1. for 0 ≤ j < qn+1 and k ∈ Z the intervals f j T k (In (x)) have disjoint interiors; 2. for 0 ≤ j < qn+1 and k ∈ Z the intervals f j T k (Jn (x)) cover R (at most twice). Proof. We assume for instance that n is even. Since the two statements are invariant under conjugacy, we can assume that f is the translation by α and x = 0. With the notations of Corollary 2.2, write f j T k (0) = xl ; then f j T k (In (0)) = (xl , xl + βn ) and f j T k (Jn (0)) = (xl − βn , xl + βn ). Therefore both statements in the Lemma follow from Corollary 2.2. Remark 3.8 We note that in what follows the hypothesis C 2 can be replaced by f ∈ C 1+BV , i.e. f is C 1 and Df has bounded variation. In fact we will always need control of variation of the function log Df (or of something similar) when Df is positive1 . The following proposition shows that using Lemma 3.7 we can bound the variation of log Df j on the intervals In (x) with the total variation of log Df on [0, 1]. Using Proposition 3.9 we then prove Corollaries 3.10 and 3.11, which, together with Corollary 3.12, imply Denjoy’s Theorem (Corollary 3.13). Proposition 3.9 Let f ∈ C 2 then for 0 ≤ j ≤ qn+1 Var logDf j ≤ Var logDf = V In (x) 1
(3.3)
[0,1]
If ψ has bounded variation and φ is Lipschitz with constant Lφ then Var (φ◦ψ) ≤ Lφ Var ψ
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Proof. The chain rule applied to f j gives: log Df j =
j−1
log Df ◦ f i .
(3.4)
i=0
The estimate (3.3) then follows from the ﬁrst assertion of Lemma 3.7. Proposition 3.9 has the following corollaries: Corollary 3.10 Let f, V, n and j be as in Proposition 3.9. Then In f j (x)  −V ≤ eV . ≤ e In (x)Df j (x) Proof. We ﬁrst note that In f j (x) = f j (In (x)) and secondly that
(3.5)
f j (In (x)) ≤ In (x)Df j (ξ) for some ξ ∈ In (x). Thus by (3.3) and Lemma 3.7 one has % % In f j (x)  %% %% % % j j ≤ log Df (ξ) − log Df (x) % % % ≤ V. % log In (x)Df j (x)
Corollary 3.11 Let f, V, n and j be as in Proposition 3.9. Then log Dfn ◦ f j (x) − log Dfn (x) ≤ V
(3.6)
Proof. Since fn and f j commute we can write log Dfn ◦ f j + log Df j = log Df j ◦ fn + log Dfn .
(3.7)
On the other hand, it follows from Proposition 3.9 that log Df j ◦ fn − log Df j  ≤ V.
(3.8)
Corollary 3.12 (Denjoy’s inequality) Let f and V be as in Proposition 3.9. Then log Dfn C 0 ≤ 2V
(3.9)
Proof. Since Dfn is a periodic function with mean value 1, there exists x0 ∈ R such that Dfn (x0 ) = 1, i.e. log Dfn (x0 ) = 0. Given x ∈ R, take 0 ≤ j < qn+1 and k ∈ Z such that x1 = f j T k (x0 ) ∈ Jn (x). Then we have: log Dfn (x1 ) = log Dfn (x1 ) − log Dfn (x0 ) ≤ V,
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and since either2 x1 ∈ In (x) or x ∈ In (x1 ), we also have log Dfn (x) − log Dfn (x1 ) ≤ V. We can then conclude: log Dfn (x) ≤ log Dfn (x) − log Dfn (x1 ) + log Dfn (x1 ) − log Dfn (x0 ) ≤ 2V.
Corollary 3.13 (Denjoy’s Theorem) All C 2 Z2 actions (3.2) with irrational rotation number are topologically conjugated to the standard action (2.16) by translations of (−1, α). Proof. This follows from Proposition 3.6 as Denjoy’s inequality imply that the (fn )n≥0 are equicontinuous. 3.4
Denjoy’s counterexamples
Denjoy has constructed examples of C 1 diﬀeomorphisms with no periodic orbits but not topologically conjugated to a rotation [De]. Indeed one can actually prove that Theorem 3.14 For every irrational α, there exists a diﬀeomorphism f of class C 2−ε for all ε > 0, with rotation number equal to α which is not topologically conjugated to a rotation. We can ﬁnd similar results in [DMVS,He1]. 3.5
The Schwarzian derivative
To get a more precise control of the nonlinearity of fn we introduce the Schwarzian derivative S. Let F ∈ Diﬀ 3+ (T) and f a lift of F , then the the Schwarzian derivative of f is deﬁned by 3 1 D3 f 2 Sf = D log Df − (D log Df ) = − 2 Df 2 2
D2 f Df
2 (3.10)
The following proposition collects some properties of S. Proposition 3.15 Let f ∈ Diﬀ 3+ (R) then 2
1. S (f ◦ g) = (Sf ) ◦ g (Dg) + Sg; 2 n−1 2. for n ≥ 1 then Sf n = i=0 (Sf ) ◦ f i Df i ; √ 3. D log Df C 0 ≤ 2 Sf C 0 . 2
In fact if x1 ∈ In (x) then x1 ∈ fn−1 (x) , x , from which it follows that x1 ≤ x ≤ fn (x1 ), i.e. x ∈ In (x1 ).
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Proof. 1. is immediate and 2. can be checked by induction using 1. To prove 3. choose x0 such that D log Df C 0 = D log Df (x0 ), i.e. D2 log Df (x0 ) = 0. Then D log Df (x0 ) = 2Sf (x0 ) from which 3. follows. Corollary 3.16 Let f ∈ Diﬀ 3+ (R), S = Sf C 0 and Mn = fn − idC 0 . For 0 ≤ j ≤ qn+1 one has Sf j (x) ≤
Mn e2V S In (x)2
Proof. Using 2. of Proposition 3.15 we get j−1 % 2 %% % Sf j (x) = % (Sf ) ◦ f i (x) Df i (x) % i=0
≤S
j−1
Df i (x)2
i=0
Using Corollary 3.10 we obtain: j−1 Se2V i In f (x) 2 . 2 In (x) i=0 2 Since Mn = maxx In (x), the trivial ai ≤ supi ai  ai , and i inequality the disjointness of the intervals In f (x) lead to the desired conclusion.
Sf j (x) ≤
Corollary 3.17 Let f ∈ Diﬀ 3+ (R), mn = minx In (x) and Mn = maxx In (x). For 0 ≤ j ≤ qn+1 one has 1
D log Df  j
with c1 =
C0
√ 2SeV .
c1 Mn2 ≤ mn
Proof. Using 3. of Proposition 3.15 and Corollary 3.16 we have √ 1 √ √ (Mn ) 2 eV S D log Df j C 0 ≤ 2 Sf j C 0 ≤ 2 . minx In (x) Corollary 3.18 Let f ∈ Diﬀ 3+ (R), mn = minx In (x) and Mn = maxx In (x). For 0 ≤ j ≤ 2qn+1 one has 1
D log Df C 0 j
with c2 =
√ 2SeV 1 + e2V .
c2 Mn2 ≤ mn
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Proof. We only need to consider the case qn+1 ≤ j ≤ 2qn+1 , then i = j−qn+1 , varies between 0 and qn+1 . Applying Corollary 3.17 to D log Df i ◦ fn+1 the thesis follows. Corollary 3.19 Let f ∈ Diﬀ 3+ (R), Mn = maxx In (x). For m = n, n + 1 one has 1
D log Dfm (x) ≤ where c3 =
c3 Mn2 In (x)
√ 2Se2V 1 + e2V 2 + e2V .
Proof. Let x0 ∈ R be such that mn = In (x0 ), and let x ∈ R. Choose 0 ≤ j < qn+1 , k ∈ Z, y ∈ Jn (x0 ) such that x = f j T k (y). One has D log D fm ◦ f j (y) = D log Dfm (x) Df j (y) + D log Df j (y) , and therefore, by Corollaries 3.17, 3.18: √ D log Dfm (x) ≤ 2SeV 2 + e2V Df j (y)−1 . On the other hand, from Corollary 3.10, one has Df j (y)−1 ≤
In (y) V e . In (x)
Finally, we have by Denjoy’s inequality In (fn (x0 )) ≤ e2V In (x0 ), In (fn−1 (x0 )) ≤ e2V In (x0 ), and therefore, as In (y) ⊂ In (x0 )∪In (fn (x0 )) or In (y) ⊂ In (x0 )∪In (fn−1 (x0 )): In (y) ≤ 1 + e2V In (x0 ) which proves the required inequality. We can ﬁnally prove 1
Corollary 3.20 Let f as in Proposition 3.15. Then log Dfn C 0 ≤ c4 Mn2 . Proof. Let x0 ∈ T such that mn = In (x0 ) (so log Dfn (x0 ) = 0); from Lemma 3.7 we know that Jn (x0 ) cover the circle, so for all x ∈ T there exists 0 ≤ j ≤ qn+1 and j ∈ Jn (x0 ) such that x = f j (y). Letting 0 ≤ j ≤ 2qn+1 we can take y ∈ In (x0 ). We use the fact that log Df j (fn (y)) − log Df j (y) = log Dfn (x) − log Dfn (y)
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145
we write log Dfn (x) − log Dfn (x0 ) = log Df j (fn (y)) − log Df j (y) + log Dfn (y) − log Dfn (x0 )
(3.11) (3.12)
To estimate the right hand side of (3.11) we use log Df j (fn (y)) − log Df j (y) ≤ 
fn (y)
D log Df j (ξ) dξ
y
≤ mn (y) D log Df j C 0 ≤
1 In (y) c2 Mn2 In (x0 )
but fn (y) − y ≤ fn (y) − fn (x0 ) + x0 − y ≤ (eV + 1)mn (x0 ). For (3.12) we use something similar. And ﬁnally we obtain the result. 3.6
Partial renormalization
In this subsection we prove a ﬁrst step of every renormalisation theory for circle diﬀeomorphisms. We prove that starting with some f ∈ Diﬀ ω + (R), not necessarily near a translation, then for all > 0 we can do an appropriate number of renormalisation steps in order to end with an analytic diﬀeomorphism f which is close to a translation in the C 2 topology. Proposition 3.21 Let us consider the action π : Z2 → Diﬀ ω + (R), given by: π : e−1 → π (e−1 ) = T π : e0 → π (e0 ) = f with ρ (π) = [−1, α] and α ∈ R \ Q. Let fn = f qn ◦ T pn and (fn , fn+1 ) be a basis of the Z2 action. Let > 0, then for n large enough, there exists hn ∈ Diﬀ ω + (R) such that: hn ◦ fn ◦ h−1 n =T hn ◦ fn+1 ◦ h−1 n = Fn+1 with Fn+1 ∈ Diﬀ ω + (R), ρ (Fn+1 ) = αn+1 and D log DFn+1 C 0 < . Proof. Let x0 ∈ T be such that mn = In (x0 ) and let A be the aﬃne map such that A(x0 ) = 0, A(fn (x0 )) = −1. Set fˆm = A◦fm ◦A−1 for m = n, n+1. By Corollaries 3.17 and 3.20, we have log Dfˆm C 1 ≤ cMn1/2 . Moreover fˆn (0) = −1; therefore there exists k0 ∈ Diﬀ 2+ ([−1 , 1 ]) with k0 − 1/2 idC 2 < cMn and k0 ◦ fˆn ◦ k0−1 = T on [0, 1]. On the other hand, the
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quotient R/fˆnZ is a compact analytic 1dimensional manifold, and is therefore ˆ C ω diﬀeomorphic to the circle: there exists k1 ∈ Diﬀ ω + ([−1 , 1 ]) with k1 ◦ fn ◦ k1−1 = T on [0, 1]. 2 The composition k0 ◦ k1−1 belongs to Diﬀ 2+ (T); taking k ∈ Diﬀ ω + (T), C ω −1 close to k0 ◦ k1 , and setting k2 = k ◦ k1 , we have k2 ∈ Diﬀ + ([−1 , 1 ]) and 1/2 k2 − idC 2 < cMn . ˆ −1 It is now suﬃcient to take hn = k2 ◦A: we have hn ◦fn ◦h−1 n = k2 ◦ fn ◦k2 = −1 −1 ˆ T and Fn+1 := hn ◦ fn+1 ◦ hn = k2 ◦ fn+1 ◦ k2 , with log DFn+1 C 1 ≤ cMn1/2 .
4 4.1
Analytic case A linearization criterion
Let π : Z2 → Diﬀ ω + (R) be an action with π(e−1 ) = T , π(e0 ) = f as above. Let ∆ > 0 be such that f is holomorphic and univalent in B∆ . Proposition 4.1 The action π is C ω linearizable if and only if the set 6 L= f −n (B∆ ) n≥0
contains R in its interior. Proof. The condition is obviously necessary. Assume that it is satisﬁed. Then the component U of intL which contains R is invariant under T and satisﬁes f (U ) ⊂ U . It is simply connected by the maximum principle (applied to m f n ); therefore there exists δ > 0 and a real biholomorphism h : Bδ → U commuting with T . Then h−1 ◦ f ◦ h is real, commutes with T and sends univalently Bδ into itself. Therefore it is a translation. 4.2
Local Theorem 1.2: big strips
2 4.2.1 Scheme of the proof Let π : Z2 → Diﬀ ω + (R) be an action of Z with generators π(e−1 ) = T , π(e0 ) = f . We assume that ρ(f ) = α is a Brjuno number and that f is holomorphic and univalent in a strip B∆ with 1 ∆ > 2π B(α) + c, where c is a conveniently large universal constant. We set F0 = f , ∆0 = ∆, α0 = α. We will denote by c1 , c2 , c3 universal constants. We will construct, through a geometric construction explained in the next section, holomorphic maps Hn , Fn for n > 0 with the following properties:
• (in ) Fn ∈ Diﬀ ω + (R) commutes with T , has rotation number αn , and is holomorphic and univalent in a strip B∆n with
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1 • (iin+1 ) ∆n+1 = αn−1 ∆n − 2π log αn−1 − c1 . • (iiin+1 ) The map Hn+1 is holomorphic and univalent in the rectangle Rn = 1 { e z < 2,  m z < ∆n − 2π log αn−1 − c2 }. Moreover Hn+1 restricted to (−2, 2) is real and orientation reversing. • (ivn+1 ) For z, z ∈ Rn we have αn (m Hn+1 (z) − m Hn+1 (z )) + (m z − m z ) ≤ c3 . • (vn ) For z ∈ Rn we have 11 9 αn < e(Fn (z) − z) < αn 10 10 1 αn  m(Fn (z) − z) < 10 • (vin+1 ) For z ∈ Rn ∩ T −1 Rn Hn+1 (T z) = Fn+1 (Hn+1 (z)) + an+1 • (viin+1 ) For z ∈ Rn ∩ Fn−1 (Rn ) Hn+1 (Fn (z)) = Hn+1 (z) − 1. Let us explain why these properties imply that the action generated by T, f is linearizable. We will apply the criterion of Section 4.1. We can assume c3 ≥ c1 ≥ 1. Deﬁne, for n ≥ 0 c3 (n) = c3 (1 + αn + αn αn+1 + αn αn+1 αn+2 + . . . ) < 4c3 . Take c = c2 + 8c3 . We will show that if Imz < ∆n −
1 B(αn ) − [c2 + 2c3 (n)] 2π
then for all m ≥ 0, we have  m Fnm (z) − m z ≤ c3 (n). This allows to conclude by Proposition 4.1 (taking n = 0). Assume by contradiction that for some n ≥ 0, some z and some M > 0, we have 1 B(αn ) − [c2 + 2c3 (n)], 2π m  m Fn (z) − m z ≤ c3 (n), for 0 ≤ m < M  m z < ∆n −
 m FnM (z)
− m z > c3 (n).
(4.1) (4.2) (4.3)
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We may assume that e z ∈ [0, 1). Observe that from (4.1), (4.2) we have 1 B(αn ) − [c2 + 2c3 (n)] 2π 1 log αn−1 − (c2 + 1) < ∆n − 2π
 m Fnm (z) < ∆n −
for 0 ≤ m < M , hence by (vn )  m FnM (z) < ∆n −
1 log αn−1 − c2 . 2π
Thus, if we put km = [e Fnm (z)] for 0 ≤ m ≤ M and wm = Fnm (z) − km , we have wm ∈ Rn for all 0 ≤ m ≤ M . We have also k0 = 0 and km+1 − km ∈ {0, 1, 2} by (vn ). Also, Fn (wm ) = wm+1 + (km+1 − km ). Set um = Hn+1 (wm ) for 0 ≤ m ≤ M . By (vi)n+1 and (vii)n+1 we have k
m+1 um+1 = Fn+1
−km
(um ) + (km+1 − km )an+1 − 1
and thus km um = Fn+1 (u0 ) + km an+1 − m
for 0 ≤ m ≤ M . On the other hand, by (4.3) and (ivn+1 ), we have αn (m uM − m u0 ) > c3 (n) − c3 and therefore, as c3 (n) = c3 + αn c3 (n + 1), we get  m uM − m u0  > c3 (n + 1).
(4.4)
We have also by (ivn+1 ):  m u0  ≤ αn−1 [ m z + c3 ] 1 B(αn ) − (c2 + 2c3 (n)) + c3 ] < αn−1 [∆n − 2π using (ii) 1 B(αn+1 ) + c3 αn−1 − (c2 + 2c3 (n))αn−1 ≤ ∆n+1 + c1 αn−1 − 2π 1 B(αn+1 ) − (c2 + 2c3 (n + 1)), ≤ ∆n+1 − 2π as c1 ≤ c3 . We conclude that u0 for Fn+1 and some integer M ≤ kM satisfy the same properties (4.1), (4.2), (4.3) of z for Fn and M . But we have, by (vn ) kM < e(FnM (z) − z) + 1
0 such that, if 0 ≤ m z ≤ ∆ −
1 log α0−1 − c8 = h 2π
then 1 α0 , 10 1 α0 , Df (z) − 1 ≤ 10 1 α0 . D2 f (z) ≤ 10
f (z) − z − α0  ≤
(4.8) (4.9) (4.10)
Therefore, relation (v) in Section 4.2.1 will follow provided that c2 ≥ c8 . Consider now the Jordan domain U ⊂ B∆ whose boundary is the union of the vertical segment joining ih to −ih, the image f () and the segments joining ih to f (ih), −ih to f (−ih) (see Figure 1). By (4.8), (4.9) the curve we have described has no selfintersection. We glue the two verticallike sides of U through the holomorphic map f ; we thus obtain an abstract annular ˜ . Complex conjugation leaves U invariant and induces an Riemann surface U ˜ . Deﬁne antiholomorphic automorphism of U ∆ =
1 ˜) mod(U 2
˜ is obviously not biholomorphic to C∗ , thus ∆ < +∞). We now choose (U ˜ 1 from U ˜ onto B∆ /Z, which is orientation reversing on a biholomorphism H ˜ onto the equators of these annular domains (sending the top boundary of U ˜ the bottom one of B∆ /Z); we lift H1 to an holomorphic map H1 : U ∪ ∪ f () → B∆ which satisﬁes H1 (f (z)) = H1 (z) − 1,
(4.11)
for all z ∈ . To extend the domain of H1 we will use the Lemma 4.2 Deﬁne R = { e z < 2,  m z < ∆ −
1 log α−1 − c2 } 2π
with c2 = c8 + 1. If z ∈ R and e z ≥ 0, there exists a unique integer L ≤ 0 such that f L (z) ∈ U ∪ and  m f j (z) < ∆ −
1 log α−1 − c8 2π
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151
ih f(w)
w α
0
1
ih
Fig. 1. The construction of the domain U
for 0 ≥ j ≥ L. If z ∈ R and e z < 0, there exists a unique integer L > 0 such that f L (z) ∈ U ∪ and  m f j (z) < ∆ −
1 log α−1 − c8 2π
for 0 ≤ j ≤ L. Proof. It is clear from (4.8).
We now extend the domain of deﬁnition of H1 to U ∪ R using the previous Lemma and the relation (4.11). By the Lemma, relation (4.11) will hold as soon as z ∈ R ∩ f −1 (R). Next we will deﬁne F1 (from property (vi1 ) in Section 4.2.1): set V = H1 ((U ∪∪f ())∩R), V ∗ = ∪n∈Z T n (V ), and denote by ∆∗ the largest ∆1 such that B∆1 ⊂ V ∗ (see Figure 2). For z ∈ (U ∪∪f ())∩R, deﬁne F1 (H1 (z)) = H1 (z − 1) − a1
(4.12)
(which is possible as z − 1 ∈ R); if z ∈ ∩ R, we will have F1 (H1 (z) − 1) = F1 (H1 (f (z))) = H1 (f (z) − 1) − a1 = H1 (f (z − 1)) − a1 = H1 (z − 1) − 1 − a1 = F1 (H1 (z)) − 1. This shows that F1 extends from V to V ∗ as an holomorphic map, commuting with T . Also F1 obviously restricts to a diﬀeomorphism of the real line. One
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J.C. Yoccoz
l
R ∩U
V
V*
Fig. 2. Standard construction: glueing, uniformizing, developing on the plane
also easily checks that F1 is univalent on V ∗ , and that relation (4.12) extends to all z ∈ R ∩ T −1 (R). We now have constructed F1 and H1 , and we have checked properties (i1 ), (iii1 ), (vi1 ), (vii1 ) of Section 4.2.1. We have also seen how (v) follows from the choice of R. There remains to check (ii1 ) and (iv1 ). Both follow from minorations of moduli of annular domains and Gr¨ otzsch’s Theorem. For −h ≤ y ≤ h, denote by σy the segment joining iy to f (iy). For 0 < y < h, let Ay (resp. A+ y ) be the Jordan domain whose boundary is formed of σ0 , σy , [0, iy] and f ([0, iy]) (resp. σh , σy , [iy, ih] and f ([iy, ih])). We glue through f the vertical sides of Ay , A+ y to obtain annular Riemann . surfaces A˜y , A˜+ y In the Appendix we will show that αmodA˜y − y ≤ c9 , αmodA˜+ y − (h − y) ≤ c10 .
(4.13) (4.14)
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153
1 We obtain (ii1 ) (i.e. a bound from below for ∆∗ ) taking y0 = ∆− 2π log α−1 − c2 − 1. Then, Ay0 ⊂ U ∩ R and therefore
1 mod (V ∗ /Z) ≥ α−1 (y0 − c9 ) 2 otzsch. To obtain (iv1 ) consider z ∈ R with m z > 0. and (ii1 ) follows by Gr¨ It is suﬃcient to consider z ∈ U ∪ (with m z > 0) because of (vii1 ) and (v0 ): the point z = f L (z) given by the Lemma above satisﬁes  m z − m z  < 1. On the other hand, such a z lies on a unique segment σy with  m z − y ≤
1 . 10
From Gr¨ otzsch’s Theorem, we now have a bound from below for  m H1 (z ) because modA˜y ≥ α−1 (y − c9 ) and a bound from above because −1 (h − y − c10 ) modA˜+ y ≥α
which together prove (iv1 ). This concludes the geometric part of the proof of the Theorem. 4.3
Local Theorem 1.3: small strips
The local Theorem 1.3 for small strips is an easy consequence of the local Theorem 1.2 for big strips. Consider f ∈ Diﬀ ω + (R), commuting with T , with rotation number α ∈ B, holomorphic in a strip B∆ . We will now assume that ε(f ) = sup f (z) − z − α B∆
is small. Let η ( 1 to be chosen later. We perform a construction similar to the one described in Section 4.2.2: consider the curve made of the segments joining i(∆ − η) to −i(∆ − η), i(∆ − η) to f (i(∆ − η)) and −i(∆ − η) to f (−i(∆ − η)) and the image f ([−i(∆ − η), i(∆ − η)]); this is a Jordan curve if ε(f ) is suﬃciently small, namely ε(f ) < ε0 (∆, η, α) (use Cauchy estimates to control f ([−i(∆ − η), i(∆ − η)])).
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Next we glue through f the vertical sides of the Jordan domain U bounded ˜ . We also deﬁne by this curve, to get an annular Riemann surface U R = { e z < 2,  m z < ∆ − 2η}. As in Section 4.2.2, we can deﬁne a holomorphic map H1 on R, uniformizing ˜ , and a diﬀeomorphism F1 , commuting with T , and conjugated to T f a1 by U H1 , holomorphic on a strip B∆∗1 . The key fact is the following Lemma 4.3 Given ∆1 < α−1 ∆ and ε1 > 0, we can choose η ( 1 and ε0 such that, if we start with ε(f ) < ε0 , then we obtain F1 holomorphic on B∆1 and ε(F1 ) = sup F1 (z) − z − α1  < ε1 . B∆1
The easy proof is omitted. The local Theorem 1.3 now follows from the local Theorem 1.2 by iteration −1 of the Lemma 4.3: for all N > 0, ∆N < βN −1 ∆, εN > 0, there exists ε¯0 = ε¯0 (∆N , εN , α, ∆) such that, starting with ε(f ) < ε¯0 we end up after N renormalization steps with a diﬀeomorphism FN ∈ Diﬀ ω + (R), commuting with T , holomorphic on B∆N , conjugated to T pN f qN and satisfying ε(FN ) = sup FN (z) − z − αN  < εN . B∆N
Actually, by Cauchy estimates, we can as well assume that FN is univalent on B∆N . But then, if N is large enough, we can apply local Theorem 1.2 to FN : indeed we have, if
βn−1 log αn−1
0, there exists n and a diﬀeomorphism k ∈ Diﬀ ω + (R) such that k ◦ fn ◦ k −1 = T k ◦ fn+1 ◦ k −1 = Fn+1 where Fn+1 is holomorphic and univalent on B∆0 , with rotation number αn+1 . be a small parameter to be Proof. We may assume ∆0 > 1. Let η ( ∆−1 0 determined later. By Proposition 3.21, we can ﬁnd an analytic conjugacy for the action sending (for some m = m(η, ∆0 , f )) fm to T and fm+1 to a diﬀeomorphism F with sup D log DF (x) < η∆−1 0 . R
Therefore, we can as well assume that f = F satisﬁes this estimate. We then pick ∆ > 0 such that f is holomorphic and univalent on B ∆ with τ := sup D log DF (z) < 2η∆−1 0 . B∆
Then ∆0 log 2 1 − > 2τ 2 8η provided that η is small enough. We choose n large enough to have also ∆0 ∆ > 2Mn 8η
and Mn1/2 < η.
Then the estimates in the proposition hold for w0  ≤ (8η)−1 ∆0 . In particular, taking η < 1/24 and w0 ≤ 3∆0 , we will have, for j ≤ qn+1 : % % % % %log dwj % ≤ 14η, (4.16) % dw0 % % % % % %log wj % ≤ 14η. (4.17) % w0 % and also, by Corollary 3.20
% % % % %log In (fn (x)) % < c4 η, % In (x) % % % % % %log In (fn+1 (x)) % < c4 η. % % In (x)
(4.18) (4.19)
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157
We now perform the same geometric construction as in Section 4.2 or 4.3: set h = 2∆0 In (0); for η ( ∆−1 0 , consider the Jordan domain U whose “vertical” sides are the segment [−ih, ih] and its image by fn , and horizontal sides are [−ih, fn (−ih)], [ih, fn (ih)]; glue the vertical sides of U through f ; uniformize the resulting annular Riemann surface to get an holomorphic map k : U → C which is real and orientationreversing on U ∩ R, extends continuously to the vertical sides of U , and satisﬁes k(fn (it)) = k(it) − 1
(4.20)
for t < h. Next, deﬁning the rectangle 7 8 3 R =  e z ≤ 3In (0),  m z ≤ h , 4 we use (4.16) and (4.18) (with η ( ∆−1 0 ) to extend through relation (4.20) the domain of k to U ∪ R. Setting 8 7 2 U1 = z ∈ U,  m z < h , 3 we deduce from (4.16), (4.18) that fn+1 (U1 ) ⊂ R and this allows to deﬁne a holomorphic map Fn+1 on k(U1 ) by Fn+1 (k(z)) = k(fn+1 (z)), z ∈ U1 . We will be ﬁnished if the strip B∆0 is contained in the union ∪m∈Z T m (k(U1 )). But this will be clearly true if η is small enough. 4.5
Global Theorem: proof of linearization
4.5.1 The geometric construction with large strip but small rotation number Let π : Z2 → Diﬀ ω + (R) be an analytic action with π(e−1 ) = T , π(e0 ) = f , ρ(π) = [−1, α], α ∈ (0, 1). We assume that f is holomorphic and univalent in a strip B∆ , with ∆ ≥ ∆min , ∆min a universal constant to be determined later. We also assume that α is so small that ∆≤
1 log α−1 + c0 , 2π
(4.21)
with c0 another universal constant determined as follows: if on the opposite 1 ∆ > 2π log α−1 + c0 , we are able to perform the construction of Section 4.2.2 with the estimates indicated in Section 4.2.1.
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From relations (4.5), (4.6) and (4.7) we get, for  m z ≤ 1: f (z) − z − α ≤ c5 e−2π∆ ,
(4.22)
c6 e−2π∆ , c7 e−2π∆ ,
(4.23)
Df (z) − 1 ≤ D log Df (z) ≤
(4.24)
and then, joining (4.21) with (4.22) f (z) − z ≤ c11 e−2π∆ .
(4.25)
We apply Proposition 4.4 to f in the strip B1 with n = 0. We have 1 log 2 1 wmax = min − ≥ c12 e2π∆ . , 2M0 2τ 2 Let h = θe2π∆ f (0), with θ a universal constant conveniently small, in particular θ < 12 c12 . If we have also c6 θ ( 1, then we can deﬁne a Jordan domain U whose boundary is formed of [−ih, ih], its image under f , and the segments [−ih, f (−ih)], [ih, f (ih)]. We glue the vertical sides of U through f , and uniformize the resulting Riemann surface to get a map H : U → C, real and orientationreversing on U ∩ R, which extends continuously to the vertical sides of U and satisﬁes H(f (z)) = H(z) − 1, z ∈ .
(4.26)
For small θ, we have from (4.23) 1 f (0) (4.27) 10 if  e z ≤ 3f (0),  m z ≤ h. This allows to extend the domain of H, through (4.26), to 8 7 3 R =  e z ≤ 3f (0),  m z ≤ h . 4 f (z) − z − f (0) ≤
Now, Proposition 4.4 shows that (provided θ is small) if z ∈ U and  m z ≤ 1 a1 2 h, then T f (z) ∈ R. Therefore we can deﬁne F on V = H(U ∩ Bh/2 ) by F (H(z)) = H(T f a1 (z)), z ∈ U ∩ Bh/2 .
(4.28)
Then F will extend to an holomorphic and univalent map on ∪m∈Z T (V ), commuting with T . It remains to estimate the witdth of the largest strip B∆∗ contained in ∪m∈Z T m (V ). By the Grotzsch’s Theorem mentioned earlier, this amounts to obtain a bound from below for the modulus of the annular Riemann surface V˜ obtained by glueing through f the vertical sides of V . In the appendix, we will show that m
modV˜ > c13 θe2π∆ which gives ﬁnally (with now a ﬁxed value of θ) ∆∗ ≥ c14 e2π∆ .
(4.29)
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159
4.5.2 Conclusion of the proof of the global theorem The proof of the global theorem is now easy from the above estimates and the local theorem. Recall from Section 2.5 the deﬁnition (2.11) of the functions rα (α ∈ (0, 1)) involved in the statement of conditon H. On the other side, we deﬁne, in connection with 4.2.1 (ii) and (4.29), for α ∈ (0, 1), ∆ ∈ R
1 1 if ∆ ≥ 2π log α−1 − c1 log α−1 + c0 , α−1 ∆ − 2π ∗ (4.30) rα (∆) = 1 c14 e2π∆ if ∆ < 2π log α−1 + c0 , where c0 has been chosen such that c0 > c1 + 1. ∗
Lemma 4.6 For any c > 0, there exists ∆0 such that, for any α ∈ (0, 1), ∗ any ∆∗0 ≥ ∆0 , any k ≥ 0, one has 1 rα ◦ · · · ◦ rα0 (0). 2π k−1 Proof. Left as an exercise to the reader. rα∗ k−1 ◦ · · · ◦ rα∗ 0 (∆∗0 ) ≥ c +
Let now α ∈ H and π : Z2 → Diﬀ ω + (R) be an action with π(e−1 ) = T , π(e0 ) = f , ρ(π) = [−1, α]. With the constant c involved in the statement of the local theorem (large strips), we apply the Lemma above which gives a ∗ constant ∆0 . We then make use of Corollary 4.5: we ﬁnd N and k ∈ Diﬀ ω + (R) such that k ◦ fN −1 ◦ k −1 = T, k ◦ fN ◦ k −1 = F, with F holomorphic and univalent in B∆∗ and ρ(F ) = αN . For k > 0, we 0 now set ∗
∆∗k = rα∗ N +k−1 ◦ · · · ◦ rα∗ N (∆0 ), xk = rαN +k−1 ◦ · · · ◦ rαN (0). By the very deﬁnition of condition H, there exists K such that xK ≥ B(αN +K ) which implies, by the Lemma above 1 B(αN +K ). 2π On the other side, starting from F , the geometric constructions of Sections 4.2.2 or 4.5.1 allow to obtain a sequence of diﬀeomorphisms Fk , 0 ≤ k ≤ K such that ∆∗K ≥ c +
• the action generated by T, Fk is analytically conjugated to that generated by T, F ; • Fk is holomorphic and univalent in B∆∗k , with rotation number αN +k . But ∆∗K is large enough to apply the local theorem: the action generated by T, FK (and thus also the initial action) is analytically linearizable.
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4.6
J.C. Yoccoz
Global Theorem: Construction of nonlinearizable diﬀeomorphisms
4.6.1 Introduction In this last section, we will construct, for any α ∈ / H, analytic actions with rotation number [−1, α] which are not linearizable. Actually we get even a slightly more precise result, relating the way in which condition H fails to the width of the strips of univalence for the examples. The construction of nonlinearizable actions is based on a geometric construction analogous to, but more sophisticated than, those which were made in Section 4.2.2. It is explained in Section 4.6.4. In the next section, we will state the result of this geometric construction; in Section 4.6.3, we explain how to derive the construction of analytic nonlinearizable actions from the basic step 4.6.2. 4.6.2 Result of the geometric construction In this section, we are given an analytic action π : Z2 → Diﬀ ω + (R) with π(e−1 ) = T , π(e0 ) = f and ρ(π) = [−1, α], with α > 1. We assume that f is holomorphic and univalent in a strip B∆ . Proposition 4.7 There exist universal constants c15 , c16 , . . . , c20 such that, if ∆ > c15 we are able to construct an analytic diﬀeomorphism F , commuting with T , with rotation number α−1 and the following properties: (i)
F is holomorphic and univalent in a strip B∆ with
1 log α − c17 if α < c16 ∆ α−1 ∆ + 2π ∆ = 1 if α ≥ c16 ∆. 2π log ∆ − c18
(ii) If α ≥ c16 ∆, F has a ﬁxed point p with 0 < m p < c19 < ∆ . (iii) Assume that, for some P, Q > 0, f has a periodic orbit (modZ) O with f Q (z) − P = z, z ∈ O, 0 < max m z < 2c19 ; O
then F has a periodic orbit O with f P (z ) − Q = z , z ∈ O , 0 < max m z < λ max m z, O
where
λ=
4 −1 3α c20 ∆−1
O
if α < c16 ∆ ifα ≥ c16 ∆.
We defer the proof of this proposition to Section 4.6.4.
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161
4.6.3 Construction of nonlinearizable actions Before we construct, using Proposition 4.7 as a building block, nonlinearizable analytic diﬀeomorphisms, we need a few preliminary arithmetical remarks. For α > 1, ∆ > 0, we deﬁne, in connection to Section 4.6.2:
1 log α − c17 if ∆ > c−1 α−1 ∆ + 2π 16 α, ∗ Rα (∆) = 1 (4.31) if ∆ ≤ c−1 16 α. 2π log ∆ − c18 This is closely connected to the function (rα−1 )−1 :
α−1 x + log x + 1 if x ≥ α, (rα−1 )−1 (x) = log x if x ≤ α.
(4.32)
∗ The function Rα appears in the third of the following elementary lemmas.
Lemma 4.8 Let α be irrational and x0 ≥ 0. Assume that for all k ≥ 0 we have xk := rαk−1 ◦ · · · ◦ rα0 (x0 ) ≥ log αk−1 . Then α ∈ B and xk ≥ B(αk ) − 4 for all k ≥ 0. / H if and only if there Lemma 4.9 Let α be irrational and x∗ ≥ 0. Then α ∈ exists m and an inﬁnite set I = I(m, x∗ , α) such that, for all k ∈ I −1 . rαm+k−1 ◦ · · · ◦ rαm (x∗ ) < log αm+k
Lemma 4.10 There exists a universal constant c21 > 0 with the following property. Let α be irrational, x0 > 2πc21 , N > 0; deﬁne xk := rαk−1 ◦ · · · ◦ rα0 (x0 ), choose ∆N ≥
1 2π xN
+ c21 −
1 2π x0
and deﬁne
∗ ∗ ∆k := Rα −1 ◦ · · · ◦ R −1 (∆N ), α k
0 ≤ k ≤ N,
N −1
0 ≤ k ≤ N.
Then, for all 0 ≤ k ≤ N , we have ∆k ≥
1 1 xk − c21 − max 0, xN − ∆N . 2π 2π
All the three lemmas above are elementary computations that are left to the reader. We now select some irrational number α which does not belong to H, and some width ∆∗ ≥ 1. We set 1 log+ c16 , 2π x∗ = 2π(∆∗ + c22 ),
c22 = c18 + c21 +
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J.C. Yoccoz
and we ﬁx some integer m given by Lemma 4.9 such that the set I = I(m, x∗ , α) in this lemma is inﬁnite. We will construct nonlinearizable analytic diﬀeomorphisms with rotation number αm which are holomorphic and univalent in B∆∗ . (k) Let k ∈ I. We deﬁne diﬀeomorphisms fl , 0 < l ≤ k + 1, with rotation (k) −1 number αm+l−1 and diﬀeomorphisms Fl , 0 ≤ l ≤ k, with rotation number αm+l as follows: (k)
• fk+1 = Rα−1 ; m+k
(k)
• Fl
(k)
• fl
(k)
is obtained from fl+1 by Proposition 4.7;
= T −am+l ◦ Fl
(k)
. (k)
We choose to consider fk+1 = Rα−1
m+k
(k)
on the strip B∆(k) with ∆k+1 =
−1 c−1 16 αm+k . It then follows from Proposition 4.7 that Fl morphic and univalent on a strip B∆(k) with
(k)
k+1
(k)
and fl
are holo
l
(k) ∆l
∗ ∗ = Rα ◦ · · · ◦ Rα −1 −1 (∆k+1 ). (k)
m+l
m+k
Let xl = rαm+l−1 ◦ · · · ◦ rαm (x∗ ). Observe that we have, as k ∈ I: (k)
∆k =
1 1 log αk−1 − c22 + c21 ≥ xk + c21 − c22 2π 2π
and therefore, by Lemma 4.10 (k)
∆l
≥
1 xl − c22 . 2π
In particular, taking l = 0, we have ∆0 ≥ ∆∗ . (k)
On the other hand, by (ii) in Proposition 4.7, for k ∈ I, k ≤ k, Fk has a (k) ﬁxed point pk such that (k)
(k)
0 < m pk < c19 . We take ∆∗ > 2c20 and iterate (iii) in Proposition 4.7; we obtain that F0 (k) has a periodic orbit Ok (modZ) of ﬁxed period Qk with
(k)
0 < max m z < 2 (k)
Ok
9 10
k c19 .
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163
(∞)
Finally, we extract from (F0 )k∈I a subsequence converging to F0 : this is an analytic diﬀeomorphism, holomorphic and univalent in B∆∗ , with rotation (∞) number αm , which has, for each k ∈ I, a periodic orbit Ok (modZ) of period Qk satisfying 0 < max m z ≤ 2 (∞)
Ok
(∞)
Therefore F0
9 10
k c19 .
is not linearizable. (k)
(k)
Remark 4.11 To obtain pk from (ii) in Proposition 4.7, we need ∆k +1 ≤ −1 c−1 16 αm+k ; this is obtained by deﬁnition for k = k; when k < k we can force (k)
this inequality by taking ∆k +1 smaller if necessary. 4.6.4 Proof of the Proposition 4.7 Let f, α, ∆ be as in Proposition 4.7, with ∆ > c15 , c15 large enough. By the estimates at the beginning of Section 4.2.2, there exists c23 such that, for 0 ≤ m z ≤ ∆ − c23 , we have 1 exp[−2π(∆ − c23 − m z)], 10 1 exp[−2π(∆ − c23 − m z)], Df (z) − 1 ≤ 10 1 exp[−2π(∆ − c23 − m z)]. D2 f (z) ≤ 10
f (z) − z − α ≤
On the other hand, we choose a Zperiodic function ψ in {m z ≥ ∆−c23 +1} which satisﬁes there 1 exp[−2π(m z − ∆ + c23 − 1)], 10 1 exp[−2π(m z − ∆ + c23 − 1)], Dψ(z) ≤ 10 1 exp[−2π(m z − ∆ + c23 − 1)]. D2 ψ(z) ≤ 10 ψ(z) ≤
We also choose α ∈ C, with α − α  ≤ g(z) = z + α + ψ(z),
1 10
and set
m z ≥ ∆ − c23 + 1.
We deﬁne also g(z) = g(¯ z ),
m z ≤ −(∆ − c23 + 1).
The most obvious and easiest choices are ψ ≡ 0, α = α, but other choices are allowed and may give interesting properties.
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We deﬁne a vertical strip U as follows. The left boundary is the imaginary axis, that we view as the union of the segments [−i(∆ − c23 ), i(∆ − c23 )] = , [i(∆−c23 ), i(∆−c23 +1)] = ∗ , [i(∆−c23 +1), +i∞) = + and the symmetric ¯∗ and ¯+ . The right boundary is the union of the curves f (), g(+ ), g(¯+ ), and the two (symmetric) straight segments ∗1 , ¯∗1 which connect f (i(∆ − c23 )) to g(i(∆ − c23 + 1)) and f (−i(∆ − c23 )) to g¯(−i(∆ − c23 + 1)). The estimates on f , ψ above guarantee that the domain U is well deﬁned. We will also have to consider the square S = {0 ≤ e z ≤ 1, ∆ − c23 ≤ m z ≤ ∆ − c23 + 1}
and its symmetric S. Next, we glue together some parts of the boundary of U : we glue to f () through f , + to g(+ ) through g and ¯+ to g¯(¯+ ) through g¯. We obtain a ˜ which is a twicepunctured annulus, the two punctures Riemann surface U ˜ to ∗ ∪ ∗ , corresponding to +i∞ and −i∞ and the two components of ∂ U 1 ∗ ∗ ¯ ¯ ∪ 1 (see Figures 3 and 4). The fact that ±i∞ give rise to punctures (and not holes) is checked via moduli estimates from the appendix.
g
1
S f
∆ c 23
α
S
Fig. 3.
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165
i
R
R i
Fig. 4.
˜1 the trace in U ˜ of U \ (S ∪ S). With z ∈ U \ (S ∪ S), the Denote by U formulas z→ z − 1 if e z ≥ 1, z→ f (z − 1) if 0 ≤ e z ≤ 1,  m z < ∆ − c23 , z → g(z − 1) if 0 ≤ e z ≤ 1, m z > ∆ − c23 + 1, z → g¯(z − 1)
if
0 ≤ e z ≤ 1, m z < −(∆ − c23 + 1),
˜1 → U ˜ which is holomorphic and univalent. induce after glueing a map F˜ : U ˜ Moreover, F extends holomorphically at the punctures and leave these punc˜˜ (resp. U ˜˜ ) the Riemann surface U ˜ (resp. U ˜1 ) tures ﬁxed. We denote by U 1 with the punctures ﬁlled in. Let ˜ = modU ˜, 2∆ ˜1 , ˜ 1 = modU 2∆ ˜ :U ˜ → B ˜ be a biholomorphism which is orientationreversing and let H ∆/Z ˜ Deﬁne on the equators. Let H : U \ (∗ ∪ ∗1 ∪ ¯∗ ∪ ¯∗1 ) → C a lift of H. V = H(U \ (S ∪ S)), V ∗ = ∪n∈Z T n (V ). ˜ U ˜1 ). By Grotzsch’s theorem, setting ∆ = ∆ ˜ 1 − 1 log 4, We have V ∗ /Z = H( 2π we have B∆ ⊂ V ∗ .
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On the other side, we deﬁne an analytic diﬀeomorphism F , holomorphic and univalent on B∆ , commuting with T by the condition H(z − 1) = F (H(z)), for z ∈ U , e z > 1. We will have ρ(F ) = α−1 and we need to check the other properties of Section 4.6.2. ˜ 1 from below. Let To get the estimates 4.6.2 (i), we need to bound ∆ ω = i(∆ − c23 + 1/2); consider the sets, for r > 2 AL (r) = {2 < z − ω < r} ∩ U , AR (r) = {2 < z − (ω + α) < r} ∩ U . If ∆−c23 < α2 , the sets AL (∆−c23 ), AR (∆−c23 ) do not intersect and the trace ˜ is an annular domain A1 contained in U ˜1 and separating of their union in U ˜ the equator of U1 from one of the boundary components. We therefore have ˜ 1 ≥ modA1 ∆ and we will show in the appendix that modA1 ≥
1 log ∆ − c18 2π
provided that ∆ is large enough (∆ > c15 ). If ∆ − c23 ≥ α2 , the sets AL (α/2), AR (α/2) do not intersect and the trace ˜ is an annular domain A2 contained in U ˜1 and separating of their union in U ˜ the equator of U1 from one of the boundary components. We will show in the appendix that we have modA2 ≥
1 log α − c17 . 2π
˜1 When moreover ∆ − c23 ≥ α, we also consider the annular domain A2 ⊂ U whose trace on U is the domain between the segments [0, f (0)] and [ω , f (ω 1 1 )], with ω1 = i ∆ − c23 − α2 . This annular domain is disjoint from A2 and ˜1 from the same boundary component (associated separates the equator of U to S) than A2 . In the appendix, we will show that modA2 ≥
∆ − c17 . α
We will therefore have ˜ 1 ≥ 1 log α − c17 , ∆ 2π
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if ∆ − c23 ≥ α/2, and even ˜ 1 ≥ 1 log α + α−1 ∆ − c − c , ∆ 17 17 2π when ∆ − c23 ≥ α. ˜ 1 give (i) in 4.6.2. We turn now to (ii). The two The three estimates for ∆ symmetric ﬁxed points we are interested in (when c16 ∆ ≤ α) are H(±i∞) = ˜ : in U ˜ we take out {p− , p+ }, p∓ . We deﬁne the following annular domain in U the trace of the line {e z = α/2} and the trace of the segment [−iω, iω]; this is an annular domain A3 which separates {p+ , p− } from the boundary ˜ . In the appendix we will show that components of U modA3 ≥
1 log(α∆−1 ) − c19 ≥ c19 > 0. 2π
˜ of U ˜ (ﬁlledin at ±i∞), we obtain (ii) from Considering the uniformization H standard Teichm¨ uller estimates (see [Ah]). Finally, we consider property (iii) of 4.6.2. We will assume c15 > 2c19 + c23 + 1. Let O be a periodic orbit as in 4.6.2 (iii); let z0 ∈ O with 0 ≤ e z0 < 1. We deﬁne inductively zi , i ≥ 0, by zi+1 = zi − 1 if e zi ≥ 0, zi+1 = f (zi ) if e zi < 0. Then we have zP +Q = f Q (z0 ) − P = z0 . On the other hand, let zi = H(zi ); we have zi+1 = zi − 1 if zi+1 = f (zi ), zi+1 = F (zi ) if zi+1 = zi − 1,
hence zP +Q = f P (z0 ) − Q = z0 . Because H is orientation reversing on the equators, the periodic orbit O in 4.6.2 (iii) will actually be the orbit of z¯0 . To estimate m zi , 0 ≤ i < P + Q, we consider H in the ball B of center xi = e zi and radius 12 (∆ − c23 ); the functional equation H(f (z)) = H(z)−1 allows to deﬁne H in this ball and to check that it is univalent there. In 4.6.2, we take c15 large enough and c16 small enough. If ∆ ≤ c−1 16 α, the image by H of the interval B ∩ R of length ∆ − c23 by the functional equation will be of length at most 2c−1 16 ; by Koebe’s 1/4theorem, we will have DH(xi ) ≤ c20 ∆−1 , and by the standard distorsion estimates we will obtain  m zi  < c20 ∆−1 m zi . If ∆ > c−1 16 α, we use H(f (xi )) = H(xi ) − 1 and the standard distorsion estimates to obtain DH(xi ) ≤ 54 α−1 (with c16 small enough), and then (with c15 large enough)  m zi  < 43 α−1 m zi . This concludes the proof of Proposition 4.6.2 and of the construction of non linearizable analytic diﬀeomorphisms.
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Appendix: Estimates of moduli of annular domains
5.1
Dirichlet integrals
Let A be an annular Riemann surface with boundary ∂A = ∂0 A ∂1 A. Let E(A) be the space of functions H ∈ W 1,2 (A) such that H∂0 A ≡ 0, H∂1 A ≡ 1. For H ∈ E(A) deﬁne the Dirichlet integral D(H) = ∇H2 . A
Then, we have (modA)−1 = inf D(H). E(A)
In particular, for any H0 ∈ E(A) modA ≥ D(H0 )−1 . The annular Riemann surfaces for which we need to bound from below the modulus were obtained in two similar but slightly distincts ways. 5.2
First kind of moduli estimates
In the ﬁrst setting, we have an holomorphic map f (z) = z + ϕ(z). A domain U has for boundary a vertical segment = [ih0 , ih1 ], its image f () and the segments [ih0 , f (ih0 )], [ih1 , f (ih1 )]. We obtain an annular Riemann surface A when we glue and f () through f . For t ∈ [0, 1], h ∈ [h0 , h1 ], set x + iy = ih + tϕ(ih), ϕ1 (h) = e ϕ(ih), ϕ2 (h) = m ϕ(ih). One has dx = ϕ1 (h)dt + tDϕ1 (h)dh dy = ϕ2 (h)dt + (1 + tDϕ2 (h))dh, hence dh = [ϕ1 (h)dy − ϕ2 (h)dx][ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]−1 , dx ∧ dy = [ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]dt ∧ dh. Set now H(x, y) =
h − h0 . h1 − h0
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We will have H ∈ E(A) and D(H) = (h1 − h0 )−2
∇h2 (5.1) = (h1 − h0 )−2 (ϕ21 + ϕ22 )[ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]−1 dtdh (5.2) −2
≤ (h1 − h0 )
h1
h0
5.2.1
ϕ(ih)2 dh . ϕ1 (h) − ϕ1 (h)Dϕ2 (h) − ϕ2 (h)Dϕ1 (h)
(5.3)
˜y , A ˜+ in 4.2.2 We have there, with Estimates for A y hmax = ∆ −
1 log α0−1 − c8 2π
the following estimates for 0 ≤ h ≤ hmax : ϕ(ih) − α0  ≤ c5 α0 exp(−2πc8 ) exp[−2π(hmax − h)], Dϕ(ih) ≤ c6 α0 exp(−2πc8 ) exp[−2π(hmax − h)]. Plugging in these estimates in (5.1), we obtain D(H) ≤ (h1 − h0 )−1 α0 + c9 (h1 − h0 )−2 α0 and then (D(H))−1 ≥ α0−1 (h1 − h0 )[1 − c9 (h1 − h0 )−1 ] which gives the required estimates for the moduli of A˜y , A˜+ y. 5.2.2 Estimate for V˜ in 4.5.1 One takes h0 = 0, h1 = 13 θ exp(2π∆)f (0); one has f (z) − z − f (0) ≤
1 f (0), 10
(4.27)
for  e z ≤ 3f (0),  m z ≤ h, and Df (z) − 1 ≤ c6 exp(−2π∆)
(4.23)
for  m z ≤ 1. Plugging in these estimates in (5.1) gives the required estimate: modV˜ ≥ c13 θ exp(2π∆).
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˜ in 4.6.4 The glueing there is 5.2.3 Estimates for the punctures of U made through g(z) = z + α + ψ(z), for z = ih, h ≥ ∆ − c23 + 1, with 1 exp[−2π(m z − ∆ + c23 − 1)], 10 1 exp[−2π(m z − ∆ + c23 − 1)]. Dψ(z) ≤ 10 ψ(z) ≤
From (5.1), we immediately get that the corresponding annulus has inﬁnite modulus. 5.2.4 Estimate for the modulus of A2 in 4.6.4 The glueing is made through f , which satisﬁes 1 exp[−2π(∆ − c23 − m z)], 10 1 exp[−2π(∆ − c23 − m z)], Df (z) − 1 ≤ 10
f (z) − z − α ≤
with h0 = 0 and h1 = ∆ − c23 − α2 . It is assumed that ∆ > c15 , ∆ − c23 ≥ α. We obtain in (5.1): D(H) ≤ (h1 − h0 )−2 [α(h1 − h0 ) + c˜17 ] which gives the required estimate for A2 : modA2 ≥ ∆α−1 − c17 . 5.3
Second kind of moduli estimates
The second setting is the following. On one side, we have an exactly semicircular domain AL = {r0 < z − ω < r1 , e(z − ω) > 0} and on the other an approximately semicircular domain AR , whose boundary is made from • the two arcs f ± ([ω ± ir0 , ω ± ir1 ]); • the two left halfcircles with diameters [f − (ω − irj ), f + (ω + irj )], j = 0, 1; the maps f − , f + here are close to a translation by some real number α > 1. We glue the “vertical” sides of AL , AR through f + , f − to obtain A. For r0 ≤ r ≤ r1 , 0 ≤ t ≤ 12 , we set z(t, r) = x(t, r) + iy(t, r) = ω +
1 exp(2πit)r. i
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On the other hand, for 1/2 ≤ t ≤ 1, r0 ≤ r ≤ r1 , we set 1 z(t, r) = x(t, r) + iy(t, r) = fc (r) + ρ(r) exp(2πit), i with 1 − [f (ω − ir) + f + (ω + ir)] 2 1 ρ(r) = [f + (ω + ir) − f − (ω − ir)]. 2i
fc (r) =
The formula log r/r0 log r1 /r0
H(z(t, r)) = deﬁnes a function in E(A), with
−2 r1 D(H) = log ∇(log r)2 r0 A r1 r1 dr 2 = π log . ∇(log r) = π r r 0 AL r0 In AR , we write dx + idy = adr + bdt with 1 a = a0 + ia1 = Dfc (r) + Dρ(r) exp(2πit), i b = b0 + ib1 = 2πρ(r) exp(2πit), giving
1
∇(log r)2 = AR
1/2
r
r0
b2 dtdr. a0 b1 − a1 b0 
Now, we have 5−1 4 Dfc  Dρ b2 − ≤ 2π e a0 b1 − a1 b0  ρ ρ giving
∇(log r) ≤ π 2
AR
r1
r0
4
e
Dρ ρ
Dfc  − ρ
5−1 dr.
(5.4)
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5.3.1 Estimates for A1 , A2 in 4.6.4 We take f + = g, f − = f , r0 = 3, ω = i ∆ − c23 + 12 and
∆ − c23 − 1 if 5 ≤ ∆ − c23 < α2 r1 = α if ∆ − c23 ≥ α2 ≥ 5. 2 −1 We have then, from the estimates on f , g in 4.6.4 % % 4 % % %ρ(r) − α − α + r % ≤ 1 exp −2π r − % % 10 2i 4 1 Dρ(r) − 1 ≤ exp −2π r − 10 4 1 Dfc (r) ≤ exp −2π r − 10
5 1 , 2 5 1 , 2 5 1 . 2
When we plug these estimates in (5.4), we obtain r1 ∇(log r)2 ≤ π log + c18 r0 AR giving the required estimates for modA1 , modA2 in 4.6.4. 5.3.2 Estimate for A3 in 4.6.4 We take now (with c16 ∆ ≤ α, c16 small) ω = 0, f + = g, f − = g¯, r0 = ∆ − c23 + 2, r1 = α2 − 1. The same computation as above will give r1 ∇(log r)2 ≤ π log + c18 r 0 AR and the required estimate for modA3 . Acknowledgements. I am extremely grateful to Timoteo Carletti, who wrote
a ﬁrst version of these notes, which served as a basis to the present manuscript. I would also like to thank very much Stefano Marmi, whose friendly pressure and constant help ﬁnally allowed this text to exist.
References [Ah]
L.V. Ahlfors: Lectures on Quasiconformal Mappings, Van Nostrand (1966) [Ar1] V.I. Arnol’d: Small denominators I. On the mapping of a circle into itself. Izv. Akad. Nauk. Math. Series 25, 21–86 (1961) [Transl. A.M.S. Serie 2, 46, (1965)] [DMVS] W. De Melo and S.J. Van Strien: Onedimensional dynamics, SpringerVerlag, New York, Heidelberg, Berlin (1993)
Analytic linearization of circle diﬀeomorphisms [De]
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A. Denjoy: Sur les courbes d´eﬁnies par les ´equations diﬀerentielles ` a la surface du tore, J. Math. Pures et Appl. 11, s´erie 9, 333–375, (1932) [HW] G.H. Hardy and E.M. Wright: An introduction to the theory of numbers, 5th edition Oxford Univ. Press [He1] M.R. Herman: Sur la conjugaison diﬀerentielle des diﬀ´eomorphismes du cercle a ` des rotations. Publ.Math. I.H.E.S. 49, 5–234 (1979) [He2] M.R. Herman: Simple proofs of local conjugacy theorems for diﬀeomorphisms of the circle with almost every rotation number, Bol. Soc. Bras. Mat. 16, 45–83 (1985) [KO1] Y. Katznelson and D. Ornstein: The diﬀerentiability of the conjugation of certain diﬀeomorphisms of the circle, Ergod. Th. and Dyn. Sys. 9, 643–680 (1989) [KO2] Y.Katznelson and D.Ornstein: The absolute continuity of the conjugation of certain diﬀeomorphisms of the circle, Ergod. Th. and Dyn. Sys. 9, 681– 690 (1989) [KS] K.M. Khanin and Ya. Sinai: A new proof of M. Herman’s theorem, Commun. Math. Phys. 112, 89–101 (1987) [MMY1] S. Marmi, P. Moussa and J.C. Yoccoz: The Brjuno functions and their regularity properties, Communications in Mathematical Physics 186, 265–293 (1997) [MMY2] S. Marmi, P. Moussa and J.C. Yoccoz: Complex Brjuno functions, Journal of the American Mathematical Society, 14, 783–841 (2001) [PM] R. P´erezMarco: Fixed points and circle maps, Acta Math. 179, 243–294 (1997) [Ri] E. Risler: Lin´earisation des perturbations holomorphes des rotations et applications, M´emoires S.M.F. 77 (1999) [SK] Ya.G. Sinai and K.M. Khanin: Smoothness of conjugacies of diﬀeomorphisms of the circle with rotations, Russ. Math. Surv. 44, 69–99 (1989) [Yo1] J.C. Yoccoz: Conjugaison diﬀerentielle des diﬀ´eomorphismes du cercle dont le nombre de rotation v´eriﬁe une condition Diophantienne, Ann. Sci. Ec. Norm. Sup. 17, 333–361(1984) [Yo2] J.C. Yoccoz: Th´eor`eme de Siegel, polynˆ omes quadratiques et nombres de Brjuno, Ast´erisque 231, 3–88 (1995)
Some open problems related to small divisors S. Marmi, J.C. Yoccoz
0
Introduction
What follows is a redaction of a (memorable) three–hours–long open problem session which took place during the workshop in Cetraro. Each of the ﬁve lecturers (H. Eliasson, M. Herman, S. Kuksin, J.N. Mather and J.C. Yoccoz) spent about half an hour brieﬂy introducing some open problems. This redaction grew from the notes that the ﬁrst author took during the session: whereas we mention who suggested each of the problems of the list we give below (with the exception of the authors of this text) we are the only responsible for any mistake the reader may ﬁnd in their description or formulation. Moreover the list of references is very far from being complete and has not been updated. We tried to make this text selfcontained, but see [KH] for terminology and further information and [Yo2] for a short survey of classical results concerning small divisor problems.
1 1.1
OneDimensional Small Divisor Problems (On Holomorphic Germs and Circle Diﬀeomorphisms) Linearization of the quadratic polynomial. Size of Siegel disks
Let us consider the linearization problem for the quadratic polynomial Pλ (z) = λ(z − z 2 ) ([Yo3], Chapter II) where z ∈ C, λ = e2πiα and α ∈ C/Z. We say that Pλ is linearizable if there exists a holomorphic map tangent to the identity hλ (z) = z + O (z 2 ) such that hλ (λz) = Pλ (hλ (z)). Then hλ is unique and we will denote rλ its radius of convergence (when λ = 1 this measures the “size” of the Siegel disk of Pλ ). The second author proved the following results: (1) there exists a bounded holomorphic function U : D → C such that for all λ ∈ D, U (λ) is equal to rλ ; (2) for all λ0 ∈ S1 , U (λ) has a nontangential limit in λ0 , which is still equal to rλ0 ; (3) if λ = e2πiα , α ∈ R \ Q, Pλ is linearizable if and only if α is a Brjuno number: if (pn /qn )n≥0 denotes the sequence of the convergents of the continued ∞ fraction expansion of α then being a Brjuno number means that n=0 logqqnn+1 < +∞;
L.H. Eliasson et al.: LNM 1784, S. Marmi and J.C. Yoccoz (Eds.), pp. 175–191, 2002. c SpringerVerlag Berlin Heidelberg 2002
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(4) There exists a universal constant C1 > 0 and for all ε > 0 there exists Cε > 0 such that for all Brjuno numbers α one has (1 − ε)B(α) − Cε ≤ − log U (e2πiα ) ≤ B(α) + C1 . Problem 1.1.1 Does the function (deﬁned on the set of Brjuno numbers) α → B(α) + log U (e2πiα ) belong to L∞ (R/Z)? There is a good numerical evidence [Mar] in support to a positive answer to the following much stronger property: Problem 1.1.2 Does the function α → B(α) + log U (e2πiα ) extend to a 1/2H¨ older continuous function as α varies in R? These two problems are discussed to some extent in [MMY1], [MMY2]. For some related analytical and numerical results concerning some areapreserving maps, including the standard family, we refer to [Mar], [BPV], [MS], [Da], [BG], [CL]. 1.2
Herman rings. Diﬀerentiable conjugacy of diﬀeomorphisms of the circle
The second author [Yo5] proved that the Brjuno condition is also necessary and suﬃcient in the local conjugacy problem for analytic diﬀeomorphisms of the circle. In this case the simplest nontrivial model is provided by the z+a Blaschke products Qa,α (z) = ρα z 2 1+az . Here a ∈ (3, +∞) and ρα ∈ S1 is chosen in such a way that the rotation number of the restriction of Qa,α to S1 is exactly α. Under these assumptions Qa,α induces an orientationpreserving analytic diﬀeomorphism of S1 . Note that when a → +∞ then Qa,α (z) → e2πiα z. When α is a Brjuno number if a is large enough then Qa,α is analytically conjugated to the rotation Rα (z) = e2πiα z in a neighborhood of S1 . If α satisﬁes the more restrictive artihmetical condition H (we refer to Yoccoz’s lectures in this volume for its deﬁnition) then Qa,α is conjugated to the rotation for all a > 3. This leads to the following Problem 1.2.1 Let α be a Brjuno number not satisfying condition H: does there exist an a > 3 such that Qa,α is not analytically conjugated to the rotation Rα ? Concerning this problem Herman showed that there exists at least a Brjuno number α not satisfying H such that the answer to the previous question is positive. In general one expects to exist a maximal interval (a0 , +∞), a0 > 3, such that Qa,α is analytically conjugated to a rotation for all a ∈ (a0 , +∞) whereas Qa0 ,α is not analytically conjugated. Problem 1.2.2 How smooth is the conjugacy for a0 ? How does this smoothness depend on α?
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This leads naturally to the study of conjugacy classes of orientationpreserving diﬀeomorphisms of the circle with ﬁnitely many continuous derivatives. Here the relevant arithmetical conditions on the rotation number are of Diophantine type. Let τ ≥ 0; we denote DC(τ ) the set of irrational numbers α whose denominators qn of the continued fraction expansion satisfy qn+1 = O(qn1+τ ) for all n ≥ 0. Let T = R/Z; r, s ∈ {0, +∞, ω} ∪ [1, +∞). Let Diﬀr+ (T) be the group of r C diﬀeomorphisms1 of T which are orientationpreserving. We denote Dr (T) the group of C r diﬀeomorphisms of the real line such that f˜−id is Zperiodic. We consider the linearization problem f ◦ h = h ◦ Rα where Rα denotes the rotation of α on T, α is the rotation number of f (mod 1) and h ∈ Diﬀs+ (T), with r ≥ s. One must distinguish the local conjugacy problem from the global one: thus we deﬁne Cr,s = {α ∈ R \ Q , every f ∈ Diﬀr+ (T) with rotation number α mod 1 is conjugated to Rα with a conjugacy h ∈ Diﬀs+ (T)} loc Cr,s = {α ∈ R \ Q , every f ∈ Diﬀr+ (T) with rotation number α mod 1
C r close to Rα is conjugated to Rα with a conjugacy h ∈ Diﬀs+ (T)} loc the neighborhood in the C r topology which Note that in the deﬁnition of Cr,s measures the distance of f from Rα depends on α. Let s < r − 1 < ∞. The following inclusions are known after [He2, KO1, KO2, KS, Yo1, Yo4], etc. loc DC(r − s − 1 − ε) ⊂ Cr,s ⊂ Cr,s ⊂ DC(r − s − 1 + ε) for all ε > 0
The third inclusion is due to Herman [He2]. One also knows from [SK] that: • if 1 < s < 2 < s + 1 < r < 3 then DC(r − s − 1) ⊂ Cr,s ; • DC(0) ⊂ Cr,r−1 provided that r > 2, r ∈ / N. loc Problem 1.2.3 Determine Cr,s and Cr,s . Are they diﬀerent ?
1.3
Gevrey classes
In the case of the conjugacy of germs of formal diﬀeomorphisms of (C, 0) one can consider a problem similar to the local one above requiring the formal germs to belong to some ultradiﬀerentiable class, for example Gevrey classes. Consider two subalgebras A1 ⊂ A2 of zC[[z]] closed with respect to the composition of formal series. In addition to the usual cases zC[[z]] (formal germs) and zC{z} (analytic germs) one can consider Gevrey for example n s classes Gs , s > 0 (i.e. series F (z) = such that there exist n≥0 fn z 1
If r = 0 it is the group of homeomorphisms of T; if r ≥ 1, r ∈ R \ N, it is the group of C [r] diﬀeomorphisms whose [r]th derivative satisﬁes a H¨ older condition of exponent r − [r]; if r = ω it is the group of Ranalytic diﬀeomorphisms.
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c1 , c2 > 0 such that fn  ≤ c1 cn2 (n!)s for all n ≥ 0). Let F ∈ A1 being such that F (0) = λ ∈ C∗ . We say that F is linearizable in A2 if there exists H ∈ A2 tangent to the identity and such that F ◦ H = H ◦ Rλ where Rλ (z) = λz. Let λ = e2πiα with α ∈ R \ Q. One knows that if α is a Brjuno number then for all s > 0 all germs F ∈ Gs have a linearization H ∈ Gs (see [CM]). Let r > s > 0 and denote Gr,s = {α ∈ R\Q , every F ∈ Gs is conjugated to Rα with a conjugacyH ∈ Gr } One knows that a condition weaker than Brjuno is suﬃcient [CM]. Problem 1.3.1 Determine Gr,s . Of course one can ask a similar question in the circle case, distinguishing the local from the global case.
2 2.1
FiniteDimensional Small Divisor Problems Linearization of germs of holomorphic diﬀeomorphisms of (Cn , 0)
Let n ≥ 2 and let f ∈ (C[[z1 , . . . zn ]])n be a germ of formal diﬀeomorphism of (Cn , 0), z = (z1 , . . . , zn ), f (z) = Az + O(z 2 ) with A ∈ GL (n, C). Let λ1 , . . . , λn denote eigenvalues of A, k = (k1 , . . . , kn ) ∈ Nn , λk = the n k1 kn λ1 . . . λn and k = j=1 kj . Assume that the eigenvalues are all distinct. If λk − λj = 0 for all j = 1, . . . , n and k ∈ Nn , k ≥ 2 (NR) then f is formally linearizable, i.e. there exists a unique germ h of formal diﬀeomorphism of (Cn , 0), tangent to the identity, such that h−1 ◦ f ◦ h = A. Let f be Canalytic and assume that A satisﬁes (NR). For m ∈ N, m ≥ 2 let Ω(m) = inf λk − λj  . 2≤k≤m 1≤j≤n
Then Brjuno [Br] proved that if A is diagonalizable, satisﬁes (NR) and the condition ∞ 2−k log(Ω(2k+1 ))−1 < +∞ (B) k=0
then f is analytically linearizable, i.e. the formal germ h deﬁnes a germ of Canalytic diﬀeomorphism of (Cn , 0). The proof uses the classical majorant series method used by Siegel [S, St] to prove that h is Canalytic under the stronger assumption that λ1 , . . . , λn satisfy a diophantine condition. Problem 2.1.1 (M. Herman) What is the optimal arithmetical condition on the eigenvalues of the linear part which assures that f is analytically linearizable? Can one obtain it by direct majorant series method?
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It seems unlikely that condition (B) is optimal. Concerning the problem of linearization of germs of holomorphic diﬀeomorphisms near a ﬁxed point [He4] contains many other questions, most of which are still open. 2.2
Elliptic ﬁxed points and KAM theory
If one replaces the assumption of being conformal with the assumption of preserving the standard symplectic structure of R2n one can consider the following problem. Let f be a real analytic symplectic diﬀeomorphism of R2n which leaves the origin ﬁxed f (0) = 0. Let z ∈ R2n and assume that f (z) = Az + O(z 2 ), where A ∈ Sp (2n, R) is conjugated in Sp (2n, R) to rα1 × . . . × rαn , where cos 2παi sin 2παi and the vector α = (α1 , . . . , αn ) ∈ Rn satisﬁes rαi = − sin 2παi cos 2παi a diophantine condition. Herman [He6] stated the following conjecture. Problem 2.2.1 (M. Herman) Show that there exists ε0 > 0 such that in the ball z ≤ ε0 there is a set of positive Lebesgue measure of invariant Lagrangian tori. Note that here the assumption of f being real analytic is essential since in the C ∞ case the conjecture is true for n = 1, open if n = 2 but one knows the existence of counterexamples if n ≥ 3. Using Birkhoﬀ normal form one can prove that the conjecture is true in many special cases (some twist condition, see [BHS]). 2.3
Zk actions
Let k ≥ 1 and F1 , . . . , Fk be commuting diﬀeomorphisms in Dr (T). Let α1 , . . . , αk be the rotation numbers of F1 , . . . , Fk . Then the diﬀeomorphisms f1 , . . . , fk of the circle induced by F1 , . . . , Fk generate a Zk action on T. If α1 is irrational and F1 = Rα1 , then we must have Fi = Rαi for 1 ≤ i ≤ k, because the centralizer of Rα1 in D0 (T) is the group of translations. Therefore, if α1 is irrational and F1 is conjugated to Rα1 by a diﬀeomorphism h ∈ Ds (T), the full Zk action is linearized by h. J. Moser [M2] has shown that, if for some γ, τ and all q > 0 max qαi ≥ γq −τ
1≤i≤k
then there exists a neighborhood V of the identity in D∞ (T) such that if Fi ∈ D∞ (T), ρ(Fi ) = αi and Fi ◦ R−αi ∈ V then the action is linearizable in D∞ (T). The simultaneous approximation condition above is probably optimal in the C ∞ category.
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More generally, one can deﬁne, for 1 ≤ s ≤ r ≤ ∞, or r = s = ω k Cr,s = {(α1 , . . . , αk ), any kuple (F1 , . . . , Fk )
of commuting diﬀeomorphisms in (Dr (T))k with rotation numbers (α1 , . . . , αk ) is linearizable in Ds (T)}, k,loc if one assumes furthermore that Fi ◦ R−αi , 1 ≤ i ≤ k, and similarly Cr,s r are C close to the identity. k k,loc Problem 2.3.1 Determine Cr,s , Cr,s . Progress in this direction is contained in the papers [DL], [Kra], [PM], etc. The previous problem generalizes to the study of Zk actions on Rn . Let k > n n ≥ 1 and consider an action Zk → Diﬀω + (R ). Among them the actions by translations Rαi : x → x+αi , 1 ≤ i ≤ k, where αi ∈ Rn , play a distinguished role. Assume that α1 , . . . , αk generate Rn and consider those actions whose n ω generators f1 , . . . , fk ∈ Diﬀω + (R ) are C close to Rα1 , . . . , Rαk and which 0 are C conjugated to it (thus one can call α1 , . . . , αk the rotation numbers of the action. Problem 2.3.2 For which rotation numbers is this C 0 conjugacy indeed analytic?
2.4
Diﬀeomorphisms of compact manifolds
Let M be a C ∞ compact connected manifold. We denote Diﬀ∞ (M ) the group of C ∞ diﬀeomorphisms of M with the C ∞ topology and Diﬀ∞ + (M ) the group of C ∞ diﬀeomorphisms C ∞ isotopic to the identity (i.e. the connected component of Diﬀ∞ (M ) which contains the identity). The general problem that one may address is to study the structure (conjugacy classes, centralizers, etc. ) of Diﬀ∞ (M ). In the special case of the ndimensional torus Tn = Rn /Zn one has a KAM theorem which describes the n local structure of Diﬀ∞ + (T ) near diophantine translations Rα : x → x + α, n n α ∈ T : there exists a neighborhood Uα of Rα in Diﬀ∞ + (T ) such that if ∞ n n f ∈ Uα there exists λ ∈ T and g ∈ Diﬀ+ (T ) such that g(0) = 0 and f = Rλ ◦ g −1 ◦ Rα ◦ g. Moreover this decomposition is locally unique. In [He1] a “converse” of this theorem is asked: Problem 2.4.1 (M. Herman) Let V be a compact C ∞ manifold, f ∈ Diﬀ∞ (V ), U a C ∞ neighborhood of the identity, Of,U = {g ◦ f ◦ g −1 , g ∈ U }. If Of,U is a ﬁnite codimension manifold is it true that V = Tn and f is C ∞ conjugate to a Diophantine translation ? In the torus case one proves KAM theorem by means of an implicit function theorem in Fr´echet spaces (see [Ha, Bo]). The main point is that a translation Rα being diophantine is equivalent to ask that for all ϕ ∈ C ∞ (Tn ) there exist ψ ∈ C ∞ (Tn ) and λ ∈ R such that the linearized conjugacy equation ψ ◦ Rα − ψ = ϕ + λ
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holds. Then one can ask the analogue of Problem 2.4.1 for the linearized conjugacy equation Problem 2.4.2 (M. Herman) If for all ϕ ∈ C ∞ (V ) there exists ψ ∈ C ∞ (V ) and λ ∈ R such that ψ ◦ f − ψ = ϕ + λ is it true that V = Tn and f is C ∞ conjugate to a Diophantine translation ?
3 3.1
KAM Theory and Hamiltonian Systems Twist maps
We let T denote the circle R/Z and θ(mod 1) denote the standard parameter of T and x the corresponding parameter of its universal cover R. We will let y ∈ R denote the standard parameter of the second factor of T × R. Let U be an open subset of T × R which intersects each vertical line {θ} × R in an open non empty interval. We consider maps f which are diﬀeomorphisms from U onto an open subset f (U ) ⊂ T × R which also intersects each vertical line in an open non empty interval. We assume that f is orientation preserving and area preserving. Since we are in two dimensions the area preserving condition is the same as requiring that f be symplectic. Let f˜ denote the lift of f to the universal cover so that f˜(x + 1, y) = f˜(x, y) + (1, 0). We also set (x , y ) = f˜(x, y). An orientation preserving symplectic C 1 diﬀeomorphism f satisﬁes a pos itive (resp. negative) monotone twist condition if ∂x ∂y > ε (resp. < −ε) for some ﬁxed ε > 0 and for all (x, y). Geometrically this condition states that the image of a segment x =constant under f˜ forms a graph over the x axis. An integrable twist map has the form f˜(x, y) = (x + r(y), y). From the area preserving property of f it follows that y dx − ydx is a closed 1form and therefore there exists a generating function (or variational principle) h = h(x, x ) such that y = −∂1 h(x, x ) , y = ∂2 h(x, x ) . The generating function is unique up to the addition of a constant and its invariance under translations (x, x ) → (x + 1, x + 1) is equivalent to the condition that y dx − ydx is exact on T × R. Moreover, from the positive twist condition one has ∂12 h(x, x ) < 0. A rotational invariant curve is a homotopically nontrivial f invariant curve. By Birkhoﬀ’s theory (see [He3], Chapitre I), such a curve is the graph of a Lipschitz function. For near–to–integrable twist maps KAM theory provides the existence of many rotational invariant curves. Herman [He3, He5] proved that rotational invariant curves persist in twist diﬀeomorphisms which are C 3 close to an integrable map. Problem 3.1.1 (J. Mather) Does there exist an example of a C r twist area– preserving map with a rotational invariant curve which is not C 1 (separate question fo each r ∈ [1, ∞] ∪ {ω}).
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Problem 3.1.2 (J. Mather) Given a C ∞ twist diﬀeomorphism and a rotational invariant curve which is not C ∞ is it possible to destroy it by an arbitrarily small C ∞ perturbation? One knows, following [Ma3] that if the rotation number is not diophantine this is indeed possible even in the case the circle is C ∞ . [Ma3] contains also destruction results for C r twist maps (see p. 212) and [Fo] for analytic maps but in both cases there is a gap between the destruction results and the persistence results given by KAM theory. It is a classical counterexample of Arnold that there exist analytic diﬀeomorphisms of the circle, with irrational rotation number, whose conjugacy to a rigid rotation is not absolutely continuous. Since every diﬀeomorphism of the circle can be embedded as rotational invariant curve of an area–preserving monotone twist map of the annulus with the same degree of smoothness, this example, and those constructed by Denjoy [De1, De2] and Herman [He2] in the diﬀerentiable case, give examples of “regular” twist maps f having “regular” rotational invariant curves γ such that f γ is topologically conjugate to a rigid rotation but the conjugacy is not absolutely continuous. In these examples the irrational rotation number has extremely good approximations by rational numbers. The classical results of Denjoy on diﬀeomorphisms of the circle show that given an invariant curve γ, if the rotation number α of f γ is irrational then one has the following: • if f γ ∈ C 2 then f γ is topologically conjugate to Rα and every orbit is dense in γ; • there exist examples of f γ ∈ C 2−ε , ε > 0, such that no orbit is dense in γ and the limit set of the orbit of every point of γ is the same Cantor set (Denjoy minimal set). Thus even if f is smooth, limit sets diﬀerent from γ may appear provided that the invariant curve γ loses smoothness. Problem 3.1.3 (J. Mather) Does there exist a C 3 area–preserving twist map of the annulus with a rotational, not topologically transitive, invariant curve of irrational rotation number? Same problem with r ≥ 3 or even analytic. M. Herman [He3] gave an example of class C 3−ε . Hall and Trupin [HT] gave a C ∞ example without the area–preserving condition. The most important progress towards the understanding of these problems has come through the introduction of Aubry–Mather sets [AL, Ma1, Ma2, Ma4]. These are closed invariant sets given by a parametric representation x = u(θ) , y = −∂1 h(u(θ), u(θ + α)) , where u is monotone (but not necessarily continuous) and u − θ is Zperiodic. They do exist for all rotation numbers α and they are subsets of a closed Lipschitz graph. For rational α one obtains periodic orbits, whereas for irrational
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numbers one has rotational invariant curves if u is continuous (in fact Lipschitz by Birkhoﬀ’s theorem) or an invariant Cantor set if u has countably many discontinuities. In this case the Aubry–Mather set can be viewed as a Cantor set drawn on a Lipschitz graph. Another important property of Aubry–Mather sets is that the “ordering” of an orbit is the same as for the rotation by α of a circle. Mather based his proof on the variational problem 1 h(u(θ), u(θ + α))dθ 0
minimizing this functional in the class of weakly monotone functions, thus they are also called action minimising sets. Problem 3.1.4 (M. Herman) For a C r twist areapreserving diﬀeomorphism does the union of the actionminimising sets which are not closed curves and do not contain periodic orbits have Hausdorﬀ dimension 0? Here r ≥ 3; otherwise Herman himself has a counterexample. 3.2
Euler–Lagrange ﬂows
It is proved in [M1] that any monotone twist map can be obtained as the time1 map of the Hamiltonian ﬂow associated to a time–dependent, Zperiodic in time Hamiltonian H : T ∗ (T × R) × R → R satisfying Legendre condition Hyy (θ, y, t) > 0. This assures that f can also be interpolated by the time1 map of the Euler–Lagrange ﬂow associated to the Lagrangian function L : T (T × R) × R → R obtained from H by Legendre transform. More generally, let M be a a closed Riemannian manifold M and consider Lagrangians of the form kinetic energy + time periodic potential V ∈ C r (M × T) (see [Ma5] for a more general setting). Assume that M has dimension at least 2. Problem 3.2.1 (J. Mather) Is there a residual set (in the sense of Baire category) in C r (M × T) such that there exists a corresponding trajectory of the Euler–Lagrange ﬂow with kinetic energy growing to ∞ as t → ∞? Of course these systems are very far from integrable ones. De La Llave has some results concerning this problem. 3.3
nbody problem
Let n ≥ 2. Consider n + 1 point masses m0 , . . . , mn moving in an inertial reference frame R3 with the only forces acting on them being their mutual gravitational attraction. If the ith particle has position vector qi then the Newtonian equations of motion are mi q¨i = −
∂V , i = 1, . . . , n ∂qi
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Gmi mj where V (q0 , . . . , qn ) = − 0≤i<j≤n qi −q and G is the universal gravitaj tional constant. The size of the system is measured by the moment of inn ertia I = 12 i=0 mi qi 2 . The total energy of the system E = T + V = n n qi 2 angular momentum A = i=0 mi qi ∧ q˙i i=0 mi 2 +V (q0 , . . . qn ), the total n and the total momentum P = i=0 mi q˙i are integrals of the motion. We will n assume that the center of mass C = i=0 mi qi is ﬁxed at the origin, thus P = 0. J. Mather recalled a problem proposed by G.D. Birkhoﬀ [Bi]: Problem 3.3.1 (G.D. Birkhoﬀ ) Let n = 2 (three body problem) and assume that the total energy is negative. One can even assume that the three particles move on a ﬁxed plane. Is it true that the moment of inertia of the system can grow to ∞ as t → ∞ for a dense open set of initial conditions? One can ask the same question for n > 2. The restriction to negative energy is necessary since from the identity of Jacobi–Lagrange I¨ = E + T follows that if the energy E ≥ 0 all orbits which are deﬁned for all times are wandering. The only thing which is known is that even for negative energy wandering sets do exist (as Birkhoﬀ and Chazy showed long ago, see [Al]). The fact that the bounded orbits form a positive Lebesguemeasure set for the planar threebody problem is a consequence of KAM theory (see [Ar]). According to M. Herman [He6] “what seems not an unreasonable question to ask (and possibly prove in a ﬁnite time with a lot of technical details) is” Problem 3.3.2 (M. Herman) If one of the masses m0 = 1 and all the other masses mj 0 H0s (T1 ) (here H0s (T1 ) is the Sobolev space formed by zeromeanvalue functions on T1 ) contains a smooth invariant 2ndimensional manifold T 2n such that
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(a) the restriction of (KdV) to T 2n deﬁnes a Liouville–Arnold integrable Hamiltonian system; (b) T 2n ⊂ T 2m if m > n; (c) ∪T 2n is dense in each H0s (T1 ). Moreover the invariant manifolds T 2n are ﬁlled with timequasiperiodic solutions (the socalled ngap solutions, see the lectures of S. Kuksin in this volume). Also the Sine–Gordon (SG) equation under Dirichlet boundary conditions utt = uxx − sin u,
u(t, 0) = u(t, π) = 0,
(SG)
is integrable (i.e. it has ngap solutions) but the manifolds T 2n have algebraic singularities and their union is proved to be dense only near the origin. We refer to [B] and [Ku] for an introduction to the recent progress on the theory of nonlinear Hamiltonian PDEs. Problem 4.3.1 (S. Kuksin) Find a Lyapounov Theorem (not of KAM type) for Hamiltonian PDEs. I.e. prove that (under reasonable assumptions) most of timeperiodic solutions from any nondegenerate oneparameter family persist under small Hamiltonian perturbations of the equation. For further information and some progress in this direction see [Ba2]. Problem 4.3.2 (S. Kuksin) Find a nonperturbative way to obtain timeperiodic solutions of an Hamiltonian PDE (the solutions should not be travelling waves). The main object of the Lectures of Kuksin in this volume is the proof of a KAMtype theorem wich assures the persistence under small Hamiltonian perturbations of most ﬁnitegap solutions of Laxintegrable Hamiltonian PDEs like (KdV) or (SG). Nothing is known on the inﬁnitegap solutions (almost periodic solutions, see [AP] for an introduction). Following [MT] (see also [BKM]), one can ask: Problem 4.3.3 (S. Kuksin) Any Sobolev space H0m (T1 ), m ≥ 1, is ﬁlled by almost periodic solutions of (KdV). Do most of them persist under small Hamiltonian perturbations of the equation ? Presumably they do not. From the KAM theorem described in Kuksin’s lectures it also follows [BK] that most small amplitude ﬁnitegap solutions of the ϕ4 equation utt = uxx − mu + γu3 , m, γ > 0 ,
(ϕ4 )
persist under small Hamiltonian perturbations. The (ϕ4 ) equation can be indeed obtained from the (SG) equation developing sin u into Taylor series at the third order and truncating. If one sets m = 0 into (ϕ4 ) the existence of smallamplitude timequasiperiodic solutions is not known. Problem 4.3.4 (S. Kuksin) Construct small amplitude time quasiperiodic solutions for the massless ϕ4 equation utt = uxx − u3 , x ∈ T1 .
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Let us consider again the (NS) equation. As we have seen the timeﬂow preserves the Hamiltonian and the L2 norm of the solutions. One can study the following properties of the longtime behaviour of the solutions: (a) (a’) (b) (b’)
limt→∞ u(t)m = ∞ (here m is suﬃciently large); lim supt→∞ u(t)m = ∞; u(t) → 0 weakly in H s as t → ∞; u(tn ) → 0 weakly in H s for a suitable sequence tn → ∞.
Problem 4.3.5 (S. Kuksin) Which properties among a)–b’) hold for the solutions of (NS) with a typical initial condition u0 , u(0, x) = u0 (x) ∈ H m ? Note the following facts: • (a)–(b’) are all wrong for any u0 if p = 1, n = 1; • (a)–(b’) are wrong for some u0 (for any p, n) since (NS) has timeperiodic solutions; • (b) ⇒ (a) and (b’) ⇒ (a’) due to the conservation of the L2 norm. Problem 4.3.6 (S. Kuksin) Does (NS) have an invariant measure in H s such that its support (in H s ) is a set with nonempty interior (here s is suﬃciently large) ? Not that a positive answer to Problem 4.9 implies that a) is wrong almost everywhere with respect to the measure. Problem 4.3.7 (S. Kuksin) In the spirit of Nekhoroshev’s theorem one can ask the following: does an arbitrary solution of a εperturbed KdV equation remain εa close to some ﬁnite gap torus during a time interval 0 ≤ t ≤ Tε with Tε ≥ CM ε−M for all M > 0? Probably a crucial step is to prove this statement for T = ε−2 . A ﬁrst result in this direction has been obtained in [Ba1].
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